What is the general solution for sin^3(x)+sin(x)=2sin^{22}(x) ?

The general solution for sin^3(x)+sin(x)=2sin^{22}(x) is x=2pin,x=pi+2pin,x= pi/2+2pin

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Popular Trigonometry >

sin^3(x)+sin(x)=2sin^{22}(x)

Solution

sin^{3} (x) + sin (x) = 2 sin^{22} (x)

Solution

x = 2 πn, x = π + 2 πn, x = \frac{π}{2} + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

sin^{3} (x) + sin (x) = 2 sin^{22} (x)

Solve by substitution

sin^{3} (x) + sin (x) = 2 sin^{22} (x)

Let:

sin (x) = u

u^{3} + u = 2 u^{22}

u^{3} + u = 2 u^{22} : u = 0, u = 1

u^{3} + u = 2 u^{22}

Switch sides

2 u^{22} = u^{3} + u

Move

u

to the left side

2 u^{22} = u^{3} + u

Subtract

u

from both sides

2 u^{22} - u = u^{3} + u - u

Simplify

2 u^{22} - u = u^{3}

2 u^{22} - u = u^{3}

Move

u^{3}

to the left side

2 u^{22} - u = u^{3}

Subtract

u^{3}

from both sides

2 u^{22} - u - u^{3} = u^{3} - u^{3}

Simplify

2 u^{22} - u - u^{3} = 0

2 u^{22} - u - u^{3} = 0

Factor

2 u^{22} - u - u^{3} : u (u - 1) (2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1)

2 u^{22} - u - u^{3}

Factor out common term

u : u (2 u^{21} - u^{2} - 1)

2 u^{22} - u^{3} - u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{3} = u^{2} u

= 2 u^{21} u - u^{2} u - u

Factor out common term

u

= u (2 u^{21} - u^{2} - 1)

= u (2 u^{21} - u^{2} - 1)

Factor

2 u^{21} - u^{2} - 1 : (u - 1) (2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1)

2 u^{21} - u^{2} - 1

Use the rational root theorem

a_{0} = 1, a_{n} = 2

The dividers of

a_{0} : 1,

The dividers of

a_{n} : 1, 2

Therefore, check the following rational numbers:

\pm \frac{1}{1 , 2}

\frac{1}{1}

is a root of the expression, so factor out

u - 1

= (u - 1) \frac{2 u ^{21} - u ^{2} - 1}{u - 1}

\frac{2 u ^{21} - u ^{2} - 1}{u - 1} = 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1

\frac{2 u ^{21} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{21} - u ^{2} - 1}{u - 1} : \frac{2 u ^{21} - u ^{2} - 1}{u - 1} = 2 u^{20} + \frac{2 u ^{20} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{21} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{21}}{u} = 2 u^{20}

Quotient = 2 u^{20}

Multiply

u - 1

2 u^{20} : 2 u^{21} - 2 u^{20}

Subtract

2 u^{21} - 2 u^{20}

from

2 u^{21} - u^{2} - 1

to get new remainder

Remainder = 2 u^{20} - u^{2} - 1

Therefore

\frac{2 u ^{21} - u ^{2} - 1}{u - 1} = 2 u^{20} + \frac{2 u ^{20} - u ^{2} - 1}{u - 1}

= 2 u^{20} + \frac{2 u ^{20} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{20} - u ^{2} - 1}{u - 1} : \frac{2 u ^{20} - u ^{2} - 1}{u - 1} = 2 u^{19} + \frac{2 u ^{19} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{20} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{20}}{u} = 2 u^{19}

Quotient = 2 u^{19}

Multiply

u - 1

2 u^{19} : 2 u^{20} - 2 u^{19}

Subtract

2 u^{20} - 2 u^{19}

from

2 u^{20} - u^{2} - 1

to get new remainder

Remainder = 2 u^{19} - u^{2} - 1

Therefore

\frac{2 u ^{20} - u ^{2} - 1}{u - 1} = 2 u^{19} + \frac{2 u ^{19} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + \frac{2 u ^{19} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{19} - u ^{2} - 1}{u - 1} : \frac{2 u ^{19} - u ^{2} - 1}{u - 1} = 2 u^{18} + \frac{2 u ^{18} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{19} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{19}}{u} = 2 u^{18}

Quotient = 2 u^{18}

Multiply

u - 1

2 u^{18} : 2 u^{19} - 2 u^{18}

Subtract

2 u^{19} - 2 u^{18}

from

2 u^{19} - u^{2} - 1

to get new remainder

Remainder = 2 u^{18} - u^{2} - 1

Therefore

\frac{2 u ^{19} - u ^{2} - 1}{u - 1} = 2 u^{18} + \frac{2 u ^{18} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + \frac{2 u ^{18} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{18} - u ^{2} - 1}{u - 1} : \frac{2 u ^{18} - u ^{2} - 1}{u - 1} = 2 u^{17} + \frac{2 u ^{17} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{18} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{18}}{u} = 2 u^{17}

Quotient = 2 u^{17}

Multiply

u - 1

2 u^{17} : 2 u^{18} - 2 u^{17}

Subtract

2 u^{18} - 2 u^{17}

from

2 u^{18} - u^{2} - 1

to get new remainder

Remainder = 2 u^{17} - u^{2} - 1

Therefore

\frac{2 u ^{18} - u ^{2} - 1}{u - 1} = 2 u^{17} + \frac{2 u ^{17} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + \frac{2 u ^{17} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{17} - u ^{2} - 1}{u - 1} : \frac{2 u ^{17} - u ^{2} - 1}{u - 1} = 2 u^{16} + \frac{2 u ^{16} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{17} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{17}}{u} = 2 u^{16}

Quotient = 2 u^{16}

Multiply

u - 1

2 u^{16} : 2 u^{17} - 2 u^{16}

Subtract

2 u^{17} - 2 u^{16}

from

2 u^{17} - u^{2} - 1

to get new remainder

Remainder = 2 u^{16} - u^{2} - 1

Therefore

\frac{2 u ^{17} - u ^{2} - 1}{u - 1} = 2 u^{16} + \frac{2 u ^{16} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + \frac{2 u ^{16} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{16} - u ^{2} - 1}{u - 1} : \frac{2 u ^{16} - u ^{2} - 1}{u - 1} = 2 u^{15} + \frac{2 u ^{15} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{16} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{16}}{u} = 2 u^{15}

Quotient = 2 u^{15}

Multiply

u - 1

2 u^{15} : 2 u^{16} - 2 u^{15}

Subtract

2 u^{16} - 2 u^{15}

from

2 u^{16} - u^{2} - 1

to get new remainder

Remainder = 2 u^{15} - u^{2} - 1

Therefore

\frac{2 u ^{16} - u ^{2} - 1}{u - 1} = 2 u^{15} + \frac{2 u ^{15} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + \frac{2 u ^{15} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{15} - u ^{2} - 1}{u - 1} : \frac{2 u ^{15} - u ^{2} - 1}{u - 1} = 2 u^{14} + \frac{2 u ^{14} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{15} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{15}}{u} = 2 u^{14}

Quotient = 2 u^{14}

Multiply

u - 1

2 u^{14} : 2 u^{15} - 2 u^{14}

Subtract

2 u^{15} - 2 u^{14}

from

2 u^{15} - u^{2} - 1

to get new remainder

Remainder = 2 u^{14} - u^{2} - 1

Therefore

\frac{2 u ^{15} - u ^{2} - 1}{u - 1} = 2 u^{14} + \frac{2 u ^{14} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + \frac{2 u ^{14} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{14} - u ^{2} - 1}{u - 1} : \frac{2 u ^{14} - u ^{2} - 1}{u - 1} = 2 u^{13} + \frac{2 u ^{13} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{14} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{14}}{u} = 2 u^{13}

Quotient = 2 u^{13}

Multiply

u - 1

2 u^{13} : 2 u^{14} - 2 u^{13}

Subtract

2 u^{14} - 2 u^{13}

from

2 u^{14} - u^{2} - 1

to get new remainder

Remainder = 2 u^{13} - u^{2} - 1

Therefore

\frac{2 u ^{14} - u ^{2} - 1}{u - 1} = 2 u^{13} + \frac{2 u ^{13} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + \frac{2 u ^{13} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{13} - u ^{2} - 1}{u - 1} : \frac{2 u ^{13} - u ^{2} - 1}{u - 1} = 2 u^{12} + \frac{2 u ^{12} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{13} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{13}}{u} = 2 u^{12}

Quotient = 2 u^{12}

Multiply

u - 1

2 u^{12} : 2 u^{13} - 2 u^{12}

Subtract

2 u^{13} - 2 u^{12}

from

2 u^{13} - u^{2} - 1

to get new remainder

Remainder = 2 u^{12} - u^{2} - 1

Therefore

\frac{2 u ^{13} - u ^{2} - 1}{u - 1} = 2 u^{12} + \frac{2 u ^{12} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + \frac{2 u ^{12} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{12} - u ^{2} - 1}{u - 1} : \frac{2 u ^{12} - u ^{2} - 1}{u - 1} = 2 u^{11} + \frac{2 u ^{11} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{12} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{12}}{u} = 2 u^{11}

Quotient = 2 u^{11}

Multiply

u - 1

2 u^{11} : 2 u^{12} - 2 u^{11}

Subtract

2 u^{12} - 2 u^{11}

from

2 u^{12} - u^{2} - 1

to get new remainder

Remainder = 2 u^{11} - u^{2} - 1

Therefore

\frac{2 u ^{12} - u ^{2} - 1}{u - 1} = 2 u^{11} + \frac{2 u ^{11} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + \frac{2 u ^{11} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{11} - u ^{2} - 1}{u - 1} : \frac{2 u ^{11} - u ^{2} - 1}{u - 1} = 2 u^{10} + \frac{2 u ^{10} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{11} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{11}}{u} = 2 u^{10}

Quotient = 2 u^{10}

Multiply

u - 1

2 u^{10} : 2 u^{11} - 2 u^{10}

Subtract

2 u^{11} - 2 u^{10}

from

2 u^{11} - u^{2} - 1

to get new remainder

Remainder = 2 u^{10} - u^{2} - 1

Therefore

\frac{2 u ^{11} - u ^{2} - 1}{u - 1} = 2 u^{10} + \frac{2 u ^{10} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + \frac{2 u ^{10} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{10} - u ^{2} - 1}{u - 1} : \frac{2 u ^{10} - u ^{2} - 1}{u - 1} = 2 u^{9} + \frac{2 u ^{9} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{10} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{10}}{u} = 2 u^{9}

Quotient = 2 u^{9}

Multiply

u - 1

2 u^{9} : 2 u^{10} - 2 u^{9}

Subtract

2 u^{10} - 2 u^{9}

from

2 u^{10} - u^{2} - 1

to get new remainder

Remainder = 2 u^{9} - u^{2} - 1

Therefore

\frac{2 u ^{10} - u ^{2} - 1}{u - 1} = 2 u^{9} + \frac{2 u ^{9} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + \frac{2 u ^{9} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{9} - u ^{2} - 1}{u - 1} : \frac{2 u ^{9} - u ^{2} - 1}{u - 1} = 2 u^{8} + \frac{2 u ^{8} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{9} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{9}}{u} = 2 u^{8}

Quotient = 2 u^{8}

Multiply

u - 1

2 u^{8} : 2 u^{9} - 2 u^{8}

Subtract

2 u^{9} - 2 u^{8}

from

2 u^{9} - u^{2} - 1

to get new remainder

Remainder = 2 u^{8} - u^{2} - 1

Therefore

\frac{2 u ^{9} - u ^{2} - 1}{u - 1} = 2 u^{8} + \frac{2 u ^{8} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + \frac{2 u ^{8} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{8} - u ^{2} - 1}{u - 1} : \frac{2 u ^{8} - u ^{2} - 1}{u - 1} = 2 u^{7} + \frac{2 u ^{7} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{8} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{8}}{u} = 2 u^{7}

Quotient = 2 u^{7}

Multiply

u - 1

2 u^{7} : 2 u^{8} - 2 u^{7}

Subtract

2 u^{8} - 2 u^{7}

from

2 u^{8} - u^{2} - 1

to get new remainder

Remainder = 2 u^{7} - u^{2} - 1

Therefore

\frac{2 u ^{8} - u ^{2} - 1}{u - 1} = 2 u^{7} + \frac{2 u ^{7} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + \frac{2 u ^{7} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{7} - u ^{2} - 1}{u - 1} : \frac{2 u ^{7} - u ^{2} - 1}{u - 1} = 2 u^{6} + \frac{2 u ^{6} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{7} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{7}}{u} = 2 u^{6}

Quotient = 2 u^{6}

Multiply

u - 1

2 u^{6} : 2 u^{7} - 2 u^{6}

Subtract

2 u^{7} - 2 u^{6}

from

2 u^{7} - u^{2} - 1

to get new remainder

Remainder = 2 u^{6} - u^{2} - 1

Therefore

\frac{2 u ^{7} - u ^{2} - 1}{u - 1} = 2 u^{6} + \frac{2 u ^{6} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + \frac{2 u ^{6} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{6} - u ^{2} - 1}{u - 1} : \frac{2 u ^{6} - u ^{2} - 1}{u - 1} = 2 u^{5} + \frac{2 u ^{5} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{6} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{6}}{u} = 2 u^{5}

Quotient = 2 u^{5}

Multiply

u - 1

2 u^{5} : 2 u^{6} - 2 u^{5}

Subtract

2 u^{6} - 2 u^{5}

from

2 u^{6} - u^{2} - 1

to get new remainder

Remainder = 2 u^{5} - u^{2} - 1

Therefore

\frac{2 u ^{6} - u ^{2} - 1}{u - 1} = 2 u^{5} + \frac{2 u ^{5} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + \frac{2 u ^{5} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{5} - u ^{2} - 1}{u - 1} : \frac{2 u ^{5} - u ^{2} - 1}{u - 1} = 2 u^{4} + \frac{2 u ^{4} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{5} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{5}}{u} = 2 u^{4}

Quotient = 2 u^{4}

Multiply

u - 1

2 u^{4} : 2 u^{5} - 2 u^{4}

Subtract

2 u^{5} - 2 u^{4}

from

2 u^{5} - u^{2} - 1

to get new remainder

Remainder = 2 u^{4} - u^{2} - 1

Therefore

\frac{2 u ^{5} - u ^{2} - 1}{u - 1} = 2 u^{4} + \frac{2 u ^{4} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + \frac{2 u ^{4} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{4} - u ^{2} - 1}{u - 1} : \frac{2 u ^{4} - u ^{2} - 1}{u - 1} = 2 u^{3} + \frac{2 u ^{3} - u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{4} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{4}}{u} = 2 u^{3}

Quotient = 2 u^{3}

Multiply

u - 1

2 u^{3} : 2 u^{4} - 2 u^{3}

Subtract

2 u^{4} - 2 u^{3}

from

2 u^{4} - u^{2} - 1

to get new remainder

Remainder = 2 u^{3} - u^{2} - 1

Therefore

\frac{2 u ^{4} - u ^{2} - 1}{u - 1} = 2 u^{3} + \frac{2 u ^{3} - u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + \frac{2 u ^{3} - u ^{2} - 1}{u - 1}

Divide

\frac{2 u ^{3} - u ^{2} - 1}{u - 1} : \frac{2 u ^{3} - u ^{2} - 1}{u - 1} = 2 u^{2} + \frac{u ^{2} - 1}{u - 1}

Divide the leading coefficients of the numerator

2 u^{3} - u^{2} - 1

and the divisor

u - 1 : \frac{2 u ^{3}}{u} = 2 u^{2}

Quotient = 2 u^{2}

Multiply

u - 1

2 u^{2} : 2 u^{3} - 2 u^{2}

Subtract

2 u^{3} - 2 u^{2}

from

2 u^{3} - u^{2} - 1

to get new remainder

Remainder = u^{2} - 1

Therefore

\frac{2 u ^{3} - u ^{2} - 1}{u - 1} = 2 u^{2} + \frac{u ^{2} - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + \frac{u ^{2} - 1}{u - 1}

Divide

\frac{u ^{2} - 1}{u - 1} : \frac{u ^{2} - 1}{u - 1} = u + \frac{u - 1}{u - 1}

Divide the leading coefficients of the numerator

u^{2} - 1

and the divisor

u - 1 : \frac{u ^{2}}{u} = u

Quotient = u

Multiply

u - 1

u : u^{2} - u

Subtract

u^{2} - u

from

u^{2} - 1

to get new remainder

Remainder = u - 1

Therefore

\frac{u ^{2} - 1}{u - 1} = u + \frac{u - 1}{u - 1}

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + \frac{u - 1}{u - 1}

Divide

\frac{u - 1}{u - 1} : \frac{u - 1}{u - 1} = 1

Divide the leading coefficients of the numerator

u - 1

and the divisor

u - 1 : \frac{u}{u} = 1

Quotient = 1

Multiply

u - 1

1 : u - 1

Subtract

u - 1

from

u - 1

to get new remainder

Remainder = 0

Therefore

\frac{u - 1}{u - 1} = 1

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1

= 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1

= (u - 1) (2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1)

= u (u - 1) (2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1)

u (u - 1) (2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or u - 1 = 0 or 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1 = 0

Solve

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

Solve

2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1 = 0 :

No Solution for

u \in R

2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1 = 0

Find one solution for

2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1 = 0

using Newton-Raphson:

No Solution for

u \in R

2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1 = 0

Newton-Raphson Approximation Definition

f (u) = 2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1

Find

f^{^{'}} (u) : 40 u^{19} + 38 u^{18} + 36 u^{17} + 34 u^{16} + 32 u^{15} + 30 u^{14} + 28 u^{13} + 26 u^{12} + 24 u^{11} + 22 u^{10} + 20 u^{9} + 18 u^{8} + 16 u^{7} + 14 u^{6} + 12 u^{5} + 10 u^{4} + 8 u^{3} + 6 u^{2} + 4 u + 1

\frac{d}{d u} (2 u^{20} + 2 u^{19} + 2 u^{18} + 2 u^{17} + 2 u^{16} + 2 u^{15} + 2 u^{14} + 2 u^{13} + 2 u^{12} + 2 u^{11} + 2 u^{10} + 2 u^{9} + 2 u^{8} + 2 u^{7} + 2 u^{6} + 2 u^{5} + 2 u^{4} + 2 u^{3} + 2 u^{2} + u + 1)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (2 u^{20}) + \frac{d}{d u} (2 u^{19}) + \frac{d}{d u} (2 u^{18}) + \frac{d}{d u} (2 u^{17}) + \frac{d}{d u} (2 u^{16}) + \frac{d}{d u} (2 u^{15}) + \frac{d}{d u} (2 u^{14}) + \frac{d}{d u} (2 u^{13}) + \frac{d}{d u} (2 u^{12}) + \frac{d}{d u} (2 u^{11}) + \frac{d}{d u} (2 u^{10}) + \frac{d}{d u} (2 u^{9}) + \frac{d}{d u} (2 u^{8}) + \frac{d}{d u} (2 u^{7}) + \frac{d}{d u} (2 u^{6}) + \frac{d}{d u} (2 u^{5}) + \frac{d}{d u} (2 u^{4}) + \frac{d}{d u} (2 u^{3}) + \frac{d}{d u} (2 u^{2}) + \frac{d u}{d u} + \frac{d}{d u} (1)

\frac{d}{d u} (2 u^{20}) = 40 u^{19}

\frac{d}{d u} (2 u^{20})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{20})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 20 u^{20 - 1}

Simplify

= 40 u^{19}

\frac{d}{d u} (2 u^{19}) = 38 u^{18}

\frac{d}{d u} (2 u^{19})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{19})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 19 u^{19 - 1}

Simplify

= 38 u^{18}

\frac{d}{d u} (2 u^{18}) = 36 u^{17}

\frac{d}{d u} (2 u^{18})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{18})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 18 u^{18 - 1}

Simplify

= 36 u^{17}

\frac{d}{d u} (2 u^{17}) = 34 u^{16}

\frac{d}{d u} (2 u^{17})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{17})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 17 u^{17 - 1}

Simplify

= 34 u^{16}

\frac{d}{d u} (2 u^{16}) = 32 u^{15}

\frac{d}{d u} (2 u^{16})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{16})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 16 u^{16 - 1}

Simplify

= 32 u^{15}

\frac{d}{d u} (2 u^{15}) = 30 u^{14}

\frac{d}{d u} (2 u^{15})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{15})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 15 u^{15 - 1}

Simplify

= 30 u^{14}

\frac{d}{d u} (2 u^{14}) = 28 u^{13}

\frac{d}{d u} (2 u^{14})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{14})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 14 u^{14 - 1}

Simplify

= 28 u^{13}

\frac{d}{d u} (2 u^{13}) = 26 u^{12}

\frac{d}{d u} (2 u^{13})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{13})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 13 u^{13 - 1}

Simplify

= 26 u^{12}

\frac{d}{d u} (2 u^{12}) = 24 u^{11}

\frac{d}{d u} (2 u^{12})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{12})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 12 u^{12 - 1}

Simplify

= 24 u^{11}

\frac{d}{d u} (2 u^{11}) = 22 u^{10}

\frac{d}{d u} (2 u^{11})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{11})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 11 u^{11 - 1}

Simplify

= 22 u^{10}

\frac{d}{d u} (2 u^{10}) = 20 u^{9}

\frac{d}{d u} (2 u^{10})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{10})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 10 u^{10 - 1}

Simplify

= 20 u^{9}

\frac{d}{d u} (2 u^{9}) = 18 u^{8}

\frac{d}{d u} (2 u^{9})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{9})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 9 u^{9 - 1}

Simplify

= 18 u^{8}

\frac{d}{d u} (2 u^{8}) = 16 u^{7}

\frac{d}{d u} (2 u^{8})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{8})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 8 u^{8 - 1}

Simplify

= 16 u^{7}

\frac{d}{d u} (2 u^{7}) = 14 u^{6}

\frac{d}{d u} (2 u^{7})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{7})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 7 u^{7 - 1}

Simplify

= 14 u^{6}

\frac{d}{d u} (2 u^{6}) = 12 u^{5}

\frac{d}{d u} (2 u^{6})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{6})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 6 u^{6 - 1}

Simplify

= 12 u^{5}

\frac{d}{d u} (2 u^{5}) = 10 u^{4}

\frac{d}{d u} (2 u^{5})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{5})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 5 u^{5 - 1}

Simplify

= 10 u^{4}

\frac{d}{d u} (2 u^{4}) = 8 u^{3}

\frac{d}{d u} (2 u^{4})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 4 u^{4 - 1}

Simplify

= 8 u^{3}

\frac{d}{d u} (2 u^{3}) = 6 u^{2}

\frac{d}{d u} (2 u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 3 u^{3 - 1}

Simplify

= 6 u^{2}

\frac{d}{d u} (2 u^{2}) = 4 u

\frac{d}{d u} (2 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 2 u^{2 - 1}

Simplify

= 4 u

\frac{d u}{d u} = 1

\frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 1

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 40 u^{19} + 38 u^{18} + 36 u^{17} + 34 u^{16} + 32 u^{15} + 30 u^{14} + 28 u^{13} + 26 u^{12} + 24 u^{11} + 22 u^{10} + 20 u^{9} + 18 u^{8} + 16 u^{7} + 14 u^{6} + 12 u^{5} + 10 u^{4} + 8 u^{3} + 6 u^{2} + 4 u + 1 + 0

Simplify

= 40 u^{19} + 38 u^{18} + 36 u^{17} + 34 u^{16} + 32 u^{15} + 30 u^{14} + 28 u^{13} + 26 u^{12} + 24 u^{11} + 22 u^{10} + 20 u^{9} + 18 u^{8} + 16 u^{7} + 14 u^{6} + 12 u^{5} + 10 u^{4} + 8 u^{3} + 6 u^{2} + 4 u + 1

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.90476 \dots : Δ u_{1} = 0.09523 \dots

f (u_{0}) = 2 (- 1)^{20} + 2 (- 1)^{19} + 2 (- 1)^{18} + 2 (- 1)^{17} + 2 (- 1)^{16} + 2 (- 1)^{15} + 2 (- 1)^{14} + 2 (- 1)^{13} + 2 (- 1)^{12} + 2 (- 1)^{11} + 2 (- 1)^{10} + 2 (- 1)^{9} + 2 (- 1)^{8} + 2 (- 1)^{7} + 2 (- 1)^{6} + 2 (- 1)^{5} + 2 (- 1)^{4} + 2 (- 1)^{3} + 2 (- 1)^{2} + (- 1) + 1 = 2

f^{^{'}} (u_{0}) = 40 (- 1)^{19} + 38 (- 1)^{18} + 36 (- 1)^{17} + 34 (- 1)^{16} + 32 (- 1)^{15} + 30 (- 1)^{14} + 28 (- 1)^{13} + 26 (- 1)^{12} + 24 (- 1)^{11} + 22 (- 1)^{10} + 20 (- 1)^{9} + 18 (- 1)^{8} + 16 (- 1)^{7} + 14 (- 1)^{6} + 12 (- 1)^{5} + 10 (- 1)^{4} + 8 (- 1)^{3} + 6 (- 1)^{2} + 4 (- 1) + 1 = - 21

u_{1} = - 0.90476 \dots

Δ u_{1} = ∣ - 0.90476 \dots - (- 1) ∣ = 0.09523 \dots

Δ u_{1} = 0.09523 \dots

u_{2} = - 0.58245 \dots : Δ u_{2} = 0.32230 \dots

f (u_{1}) = 2 (- 0.90476 \dots)^{20} + 2 (- 0.90476 \dots)^{19} + 2 (- 0.90476 \dots)^{18} + 2 (- 0.90476 \dots)^{17} + 2 (- 0.90476 \dots)^{16} + 2 (- 0.90476 \dots)^{15} + 2 (- 0.90476 \dots)^{14} + 2 (- 0.90476 \dots)^{13} + 2 (- 0.90476 \dots)^{12} + 2 (- 0.90476 \dots)^{11} + 2 (- 0.90476 \dots)^{10} + 2 (- 0.90476 \dots)^{9} + 2 (- 0.90476 \dots)^{8} + 2 (- 0.90476 \dots)^{7} + 2 (- 0.90476 \dots)^{6} + 2 (- 0.90476 \dots)^{5} + 2 (- 0.90476 \dots)^{4} + 2 (- 0.90476 \dots)^{3} + 2 (- 0.90476 \dots)^{2} + (- 0.90476 \dots) + 1 = 1.08311 \dots

f^{^{'}} (u_{1}) = 40 (- 0.90476 \dots)^{19} + 38 (- 0.90476 \dots)^{18} + 36 (- 0.90476 \dots)^{17} + 34 (- 0.90476 \dots)^{16} + 32 (- 0.90476 \dots)^{15} + 30 (- 0.90476 \dots)^{14} + 28 (- 0.90476 \dots)^{13} + 26 (- 0.90476 \dots)^{12} + 24 (- 0.90476 \dots)^{11} + 22 (- 0.90476 \dots)^{10} + 20 (- 0.90476 \dots)^{9} + 18 (- 0.90476 \dots)^{8} + 16 (- 0.90476 \dots)^{7} + 14 (- 0.90476 \dots)^{6} + 12 (- 0.90476 \dots)^{5} + 10 (- 0.90476 \dots)^{4} + 8 (- 0.90476 \dots)^{3} + 6 (- 0.90476 \dots)^{2} + 4 (- 0.90476 \dots) + 1 = - 3.36053 \dots

u_{2} = - 0.58245 \dots

Δ u_{2} = ∣ - 0.58245 \dots - (- 0.90476 \dots) ∣ = 0.32230 \dots

Δ u_{2} = 0.32230 \dots

u_{3} = 3.61022 \dots : Δ u_{3} = 4.19268 \dots

f (u_{2}) = 2 (- 0.58245 \dots)^{20} + 2 (- 0.58245 \dots)^{19} + 2 (- 0.58245 \dots)^{18} + 2 (- 0.58245 \dots)^{17} + 2 (- 0.58245 \dots)^{16} + 2 (- 0.58245 \dots)^{15} + 2 (- 0.58245 \dots)^{14} + 2 (- 0.58245 \dots)^{13} + 2 (- 0.58245 \dots)^{12} + 2 (- 0.58245 \dots)^{11} + 2 (- 0.58245 \dots)^{10} + 2 (- 0.58245 \dots)^{9} + 2 (- 0.58245 \dots)^{8} + 2 (- 0.58245 \dots)^{7} + 2 (- 0.58245 \dots)^{6} + 2 (- 0.58245 \dots)^{5} + 2 (- 0.58245 \dots)^{4} + 2 (- 0.58245 \dots)^{3} + 2 (- 0.58245 \dots)^{2} + (- 0.58245 \dots) + 1 = 0.84632 \dots

f^{^{'}} (u_{2}) = 40 (- 0.58245 \dots)^{19} + 38 (- 0.58245 \dots)^{18} + 36 (- 0.58245 \dots)^{17} + 34 (- 0.58245 \dots)^{16} + 32 (- 0.58245 \dots)^{15} + 30 (- 0.58245 \dots)^{14} + 28 (- 0.58245 \dots)^{13} + 26 (- 0.58245 \dots)^{12} + 24 (- 0.58245 \dots)^{11} + 22 (- 0.58245 \dots)^{10} + 20 (- 0.58245 \dots)^{9} + 18 (- 0.58245 \dots)^{8} + 16 (- 0.58245 \dots)^{7} + 14 (- 0.58245 \dots)^{6} + 12 (- 0.58245 \dots)^{5} + 10 (- 0.58245 \dots)^{4} + 8 (- 0.58245 \dots)^{3} + 6 (- 0.58245 \dots)^{2} + 4 (- 0.58245 \dots) + 1 = - 0.20185 \dots

u_{3} = 3.61022 \dots

Δ u_{3} = ∣ 3.61022 \dots - (- 0.58245 \dots) ∣ = 4.19268 \dots

Δ u_{3} = 4.19268 \dots

u_{4} = 3.42618 \dots : Δ u_{4} = 0.18403 \dots

f (u_{3}) = 2 \cdot 3.61022 \dots^{20} + 2 \cdot 3.61022 \dots^{19} + 2 \cdot 3.61022 \dots^{18} + 2 \cdot 3.61022 \dots^{17} + 2 \cdot 3.61022 \dots^{16} + 2 \cdot 3.61022 \dots^{15} + 2 \cdot 3.61022 \dots^{14} + 2 \cdot 3.61022 \dots^{13} + 2 \cdot 3.61022 \dots^{12} + 2 \cdot 3.61022 \dots^{11} + 2 \cdot 3.61022 \dots^{10} + 2 \cdot 3.61022 \dots^{9} + 2 \cdot 3.61022 \dots^{8} + 2 \cdot 3.61022 \dots^{7} + 2 \cdot 3.61022 \dots^{6} + 2 \cdot 3.61022 \dots^{5} + 2 \cdot 3.61022 \dots^{4} + 2 \cdot 3.61022 \dots^{3} + 2 \cdot 3.61022 \dots^{2} + 3.61022 \dots + 1 = 391356105797.3665

f^{^{'}} (u_{3}) = 40 \cdot 3.61022 \dots^{19} + 38 \cdot 3.61022 \dots^{18} + 36 \cdot 3.61022 \dots^{17} + 34 \cdot 3.61022 \dots^{16} + 32 \cdot 3.61022 \dots^{15} + 30 \cdot 3.61022 \dots^{14} + 28 \cdot 3.61022 \dots^{13} + 26 \cdot 3.61022 \dots^{12} + 24 \cdot 3.61022 \dots^{11} + 22 \cdot 3.61022 \dots^{10} + 20 \cdot 3.61022 \dots^{9} + 18 \cdot 3.61022 \dots^{8} + 16 \cdot 3.61022 \dots^{7} + 14 \cdot 3.61022 \dots^{6} + 12 \cdot 3.61022 \dots^{5} + 10 \cdot 3.61022 \dots^{4} + 8 \cdot 3.61022 \dots^{3} + 6 \cdot 3.61022 \dots^{2} + 4 \cdot 3.61022 \dots + 1 = 2126512839249.2053

u_{4} = 3.42618 \dots

Δ u_{4} = ∣ 3.42618 \dots - 3.61022 \dots ∣ = 0.18403 \dots

Δ u_{4} = 0.18403 \dots

u_{5} = 3.25127 \dots : Δ u_{5} = 0.17491 \dots

f (u_{4}) = 2 \cdot 3.42618 \dots^{20} + 2 \cdot 3.42618 \dots^{19} + 2 \cdot 3.42618 \dots^{18} + 2 \cdot 3.42618 \dots^{17} + 2 \cdot 3.42618 \dots^{16} + 2 \cdot 3.42618 \dots^{15} + 2 \cdot 3.42618 \dots^{14} + 2 \cdot 3.42618 \dots^{13} + 2 \cdot 3.42618 \dots^{12} + 2 \cdot 3.42618 \dots^{11} + 2 \cdot 3.42618 \dots^{10} + 2 \cdot 3.42618 \dots^{9} + 2 \cdot 3.42618 \dots^{8} + 2 \cdot 3.42618 \dots^{7} + 2 \cdot 3.42618 \dots^{6} + 2 \cdot 3.42618 \dots^{5} + 2 \cdot 3.42618 \dots^{4} + 2 \cdot 3.42618 \dots^{3} + 2 \cdot 3.42618 \dots^{2} + 3.42618 \dots + 1 = 140327262334.09973

f^{^{'}} (u_{4}) = 40 \cdot 3.42618 \dots^{19} + 38 \cdot 3.42618 \dots^{18} + 36 \cdot 3.42618 \dots^{17} + 34 \cdot 3.42618 \dots^{16} + 32 \cdot 3.42618 \dots^{15} + 30 \cdot 3.42618 \dots^{14} + 28 \cdot 3.42618 \dots^{13} + 26 \cdot 3.42618 \dots^{12} + 24 \cdot 3.42618 \dots^{11} + 22 \cdot 3.42618 \dots^{10} + 20 \cdot 3.42618 \dots^{9} + 18 \cdot 3.42618 \dots^{8} + 16 \cdot 3.42618 \dots^{7} + 14 \cdot 3.42618 \dots^{6} + 12 \cdot 3.42618 \dots^{5} + 10 \cdot 3.42618 \dots^{4} + 8 \cdot 3.42618 \dots^{3} + 6 \cdot 3.42618 \dots^{2} + 4 \cdot 3.42618 \dots + 1 = 802263679492.2867

u_{5} = 3.25127 \dots

Δ u_{5} = ∣ 3.25127 \dots - 3.42618 \dots ∣ = 0.17491 \dots

Δ u_{5} = 0.17491 \dots

u_{6} = 3.08501 \dots : Δ u_{6} = 0.16625 \dots

f (u_{5}) = 2 \cdot 3.25127 \dots^{20} + 2 \cdot 3.25127 \dots^{19} + 2 \cdot 3.25127 \dots^{18} + 2 \cdot 3.25127 \dots^{17} + 2 \cdot 3.25127 \dots^{16} + 2 \cdot 3.25127 \dots^{15} + 2 \cdot 3.25127 \dots^{14} + 2 \cdot 3.25127 \dots^{13} + 2 \cdot 3.25127 \dots^{12} + 2 \cdot 3.25127 \dots^{11} + 2 \cdot 3.25127 \dots^{10} + 2 \cdot 3.25127 \dots^{9} + 2 \cdot 3.25127 \dots^{8} + 2 \cdot 3.25127 \dots^{7} + 2 \cdot 3.25127 \dots^{6} + 2 \cdot 3.25127 \dots^{5} + 2 \cdot 3.25127 \dots^{4} + 2 \cdot 3.25127 \dots^{3} + 2 \cdot 3.25127 \dots^{2} + 3.25127 \dots + 1 = 50318521009.06572

f^{^{'}} (u_{5}) = 40 \cdot 3.25127 \dots^{19} + 38 \cdot 3.25127 \dots^{18} + 36 \cdot 3.25127 \dots^{17} + 34 \cdot 3.25127 \dots^{16} + 32 \cdot 3.25127 \dots^{15} + 30 \cdot 3.25127 \dots^{14} + 28 \cdot 3.25127 \dots^{13} + 26 \cdot 3.25127 \dots^{12} + 24 \cdot 3.25127 \dots^{11} + 22 \cdot 3.25127 \dots^{10} + 20 \cdot 3.25127 \dots^{9} + 18 \cdot 3.25127 \dots^{8} + 16 \cdot 3.25127 \dots^{7} + 14 \cdot 3.25127 \dots^{6} + 12 \cdot 3.25127 \dots^{5} + 10 \cdot 3.25127 \dots^{4} + 8 \cdot 3.25127 \dots^{3} + 6 \cdot 3.25127 \dots^{2} + 4 \cdot 3.25127 \dots + 1 = 302656481865.62994

u_{6} = 3.08501 \dots

Δ u_{6} = ∣ 3.08501 \dots - 3.25127 \dots ∣ = 0.16625 \dots

Δ u_{6} = 0.16625 \dots

u_{7} = 2.92697 \dots : Δ u_{7} = 0.15804 \dots

f (u_{6}) = 2 \cdot 3.08501 \dots^{20} + 2 \cdot 3.08501 \dots^{19} + 2 \cdot 3.08501 \dots^{18} + 2 \cdot 3.08501 \dots^{17} + 2 \cdot 3.08501 \dots^{16} + 2 \cdot 3.08501 \dots^{15} + 2 \cdot 3.08501 \dots^{14} + 2 \cdot 3.08501 \dots^{13} + 2 \cdot 3.08501 \dots^{12} + 2 \cdot 3.08501 \dots^{11} + 2 \cdot 3.08501 \dots^{10} + 2 \cdot 3.08501 \dots^{9} + 2 \cdot 3.08501 \dots^{8} + 2 \cdot 3.08501 \dots^{7} + 2 \cdot 3.08501 \dots^{6} + 2 \cdot 3.08501 \dots^{5} + 2 \cdot 3.08501 \dots^{4} + 2 \cdot 3.08501 \dots^{3} + 2 \cdot 3.08501 \dots^{2} + 3.08501 \dots + 1 = 18043992829.22628

f^{^{'}} (u_{6}) = 40 \cdot 3.08501 \dots^{19} + 38 \cdot 3.08501 \dots^{18} + 36 \cdot 3.08501 \dots^{17} + 34 \cdot 3.08501 \dots^{16} + 32 \cdot 3.08501 \dots^{15} + 30 \cdot 3.08501 \dots^{14} + 28 \cdot 3.08501 \dots^{13} + 26 \cdot 3.08501 \dots^{12} + 24 \cdot 3.08501 \dots^{11} + 22 \cdot 3.08501 \dots^{10} + 20 \cdot 3.08501 \dots^{9} + 18 \cdot 3.08501 \dots^{8} + 16 \cdot 3.08501 \dots^{7} + 14 \cdot 3.08501 \dots^{6} + 12 \cdot 3.08501 \dots^{5} + 10 \cdot 3.08501 \dots^{4} + 8 \cdot 3.08501 \dots^{3} + 6 \cdot 3.08501 \dots^{2} + 4 \cdot 3.08501 \dots + 1 = 114172983680.20372

u_{7} = 2.92697 \dots

Δ u_{7} = ∣ 2.92697 \dots - 3.08501 \dots ∣ = 0.15804 \dots

Δ u_{7} = 0.15804 \dots

u_{8} = 2.77673 \dots : Δ u_{8} = 0.15024 \dots

f (u_{7}) = 2 \cdot 2.92697 \dots^{20} + 2 \cdot 2.92697 \dots^{19} + 2 \cdot 2.92697 \dots^{18} + 2 \cdot 2.92697 \dots^{17} + 2 \cdot 2.92697 \dots^{16} + 2 \cdot 2.92697 \dots^{15} + 2 \cdot 2.92697 \dots^{14} + 2 \cdot 2.92697 \dots^{13} + 2 \cdot 2.92697 \dots^{12} + 2 \cdot 2.92697 \dots^{11} + 2 \cdot 2.92697 \dots^{10} + 2 \cdot 2.92697 \dots^{9} + 2 \cdot 2.92697 \dots^{8} + 2 \cdot 2.92697 \dots^{7} + 2 \cdot 2.92697 \dots^{6} + 2 \cdot 2.92697 \dots^{5} + 2 \cdot 2.92697 \dots^{4} + 2 \cdot 2.92697 \dots^{3} + 2 \cdot 2.92697 \dots^{2} + 2.92697 \dots + 1 = 6470833347.00402

f^{^{'}} (u_{7}) = 40 \cdot 2.92697 \dots^{19} + 38 \cdot 2.92697 \dots^{18} + 36 \cdot 2.92697 \dots^{17} + 34 \cdot 2.92697 \dots^{16} + 32 \cdot 2.92697 \dots^{15} + 30 \cdot 2.92697 \dots^{14} + 28 \cdot 2.92697 \dots^{13} + 26 \cdot 2.92697 \dots^{12} + 24 \cdot 2.92697 \dots^{11} + 22 \cdot 2.92697 \dots^{10} + 20 \cdot 2.92697 \dots^{9} + 18 \cdot 2.92697 \dots^{8} + 16 \cdot 2.92697 \dots^{7} + 14 \cdot 2.92697 \dots^{6} + 12 \cdot 2.92697 \dots^{5} + 10 \cdot 2.92697 \dots^{4} + 8 \cdot 2.92697 \dots^{3} + 6 \cdot 2.92697 \dots^{2} + 4 \cdot 2.92697 \dots + 1 = 43067856733.97665

u_{8} = 2.77673 \dots

Δ u_{8} = ∣ 2.77673 \dots - 2.92697 \dots ∣ = 0.15024 \dots

Δ u_{8} = 0.15024 \dots

u_{9} = 2.63387 \dots : Δ u_{9} = 0.14285 \dots

f (u_{8}) = 2 \cdot 2.77673 \dots^{20} + 2 \cdot 2.77673 \dots^{19} + 2 \cdot 2.77673 \dots^{18} + 2 \cdot 2.77673 \dots^{17} + 2 \cdot 2.77673 \dots^{16} + 2 \cdot 2.77673 \dots^{15} + 2 \cdot 2.77673 \dots^{14} + 2 \cdot 2.77673 \dots^{13} + 2 \cdot 2.77673 \dots^{12} + 2 \cdot 2.77673 \dots^{11} + 2 \cdot 2.77673 \dots^{10} + 2 \cdot 2.77673 \dots^{9} + 2 \cdot 2.77673 \dots^{8} + 2 \cdot 2.77673 \dots^{7} + 2 \cdot 2.77673 \dots^{6} + 2 \cdot 2.77673 \dots^{5} + 2 \cdot 2.77673 \dots^{4} + 2 \cdot 2.77673 \dots^{3} + 2 \cdot 2.77673 \dots^{2} + 2.77673 \dots + 1 = 2320680563.35344 \dots

f^{^{'}} (u_{8}) = 40 \cdot 2.77673 \dots^{19} + 38 \cdot 2.77673 \dots^{18} + 36 \cdot 2.77673 \dots^{17} + 34 \cdot 2.77673 \dots^{16} + 32 \cdot 2.77673 \dots^{15} + 30 \cdot 2.77673 \dots^{14} + 28 \cdot 2.77673 \dots^{13} + 26 \cdot 2.77673 \dots^{12} + 24 \cdot 2.77673 \dots^{11} + 22 \cdot 2.77673 \dots^{10} + 20 \cdot 2.77673 \dots^{9} + 18 \cdot 2.77673 \dots^{8} + 16 \cdot 2.77673 \dots^{7} + 14 \cdot 2.77673 \dots^{6} + 12 \cdot 2.77673 \dots^{5} + 10 \cdot 2.77673 \dots^{4} + 8 \cdot 2.77673 \dots^{3} + 6 \cdot 2.77673 \dots^{2} + 4 \cdot 2.77673 \dots + 1 = 16244812495.12528

u_{9} = 2.63387 \dots

Δ u_{9} = ∣ 2.63387 \dots - 2.77673 \dots ∣ = 0.14285 \dots

Δ u_{9} = 0.14285 \dots

u_{10} = 2.49802 \dots : Δ u_{10} = 0.13585 \dots

f (u_{9}) = 2 \cdot 2.63387 \dots^{20} + 2 \cdot 2.63387 \dots^{19} + 2 \cdot 2.63387 \dots^{18} + 2 \cdot 2.63387 \dots^{17} + 2 \cdot 2.63387 \dots^{16} + 2 \cdot 2.63387 \dots^{15} + 2 \cdot 2.63387 \dots^{14} + 2 \cdot 2.63387 \dots^{13} + 2 \cdot 2.63387 \dots^{12} + 2 \cdot 2.63387 \dots^{11} + 2 \cdot 2.63387 \dots^{10} + 2 \cdot 2.63387 \dots^{9} + 2 \cdot 2.63387 \dots^{8} + 2 \cdot 2.63387 \dots^{7} + 2 \cdot 2.63387 \dots^{6} + 2 \cdot 2.63387 \dots^{5} + 2 \cdot 2.63387 \dots^{4} + 2 \cdot 2.63387 \dots^{3} + 2 \cdot 2.63387 \dots^{2} + 2.63387 \dots + 1 = 832346488.77442 \dots

f^{^{'}} (u_{9}) = 40 \cdot 2.63387 \dots^{19} + 38 \cdot 2.63387 \dots^{18} + 36 \cdot 2.63387 \dots^{17} + 34 \cdot 2.63387 \dots^{16} + 32 \cdot 2.63387 \dots^{15} + 30 \cdot 2.63387 \dots^{14} + 28 \cdot 2.63387 \dots^{13} + 26 \cdot 2.63387 \dots^{12} + 24 \cdot 2.63387 \dots^{11} + 22 \cdot 2.63387 \dots^{10} + 20 \cdot 2.63387 \dots^{9} + 18 \cdot 2.63387 \dots^{8} + 16 \cdot 2.63387 \dots^{7} + 14 \cdot 2.63387 \dots^{6} + 12 \cdot 2.63387 \dots^{5} + 10 \cdot 2.63387 \dots^{4} + 8 \cdot 2.63387 \dots^{3} + 6 \cdot 2.63387 \dots^{2} + 4 \cdot 2.63387 \dots + 1 = 6126907579.45191 \dots

u_{10} = 2.49802 \dots

Δ u_{10} = ∣ 2.49802 \dots - 2.63387 \dots ∣ = 0.13585 \dots

Δ u_{10} = 0.13585 \dots

u_{11} = 2.36880 \dots : Δ u_{11} = 0.12921 \dots

f (u_{10}) = 2 \cdot 2.49802 \dots^{20} + 2 \cdot 2.49802 \dots^{19} + 2 \cdot 2.49802 \dots^{18} + 2 \cdot 2.49802 \dots^{17} + 2 \cdot 2.49802 \dots^{16} + 2 \cdot 2.49802 \dots^{15} + 2 \cdot 2.49802 \dots^{14} + 2 \cdot 2.49802 \dots^{13} + 2 \cdot 2.49802 \dots^{12} + 2 \cdot 2.49802 \dots^{11} + 2 \cdot 2.49802 \dots^{10} + 2 \cdot 2.49802 \dots^{9} + 2 \cdot 2.49802 \dots^{8} + 2 \cdot 2.49802 \dots^{7} + 2 \cdot 2.49802 \dots^{6} + 2 \cdot 2.49802 \dots^{5} + 2 \cdot 2.49802 \dots^{4} + 2 \cdot 2.49802 \dots^{3} + 2 \cdot 2.49802 \dots^{2} + 2.49802 \dots + 1 = 298561855.74542 \dots

f^{^{'}} (u_{10}) = 40 \cdot 2.49802 \dots^{19} + 38 \cdot 2.49802 \dots^{18} + 36 \cdot 2.49802 \dots^{17} + 34 \cdot 2.49802 \dots^{16} + 32 \cdot 2.49802 \dots^{15} + 30 \cdot 2.49802 \dots^{14} + 28 \cdot 2.49802 \dots^{13} + 26 \cdot 2.49802 \dots^{12} + 24 \cdot 2.49802 \dots^{11} + 22 \cdot 2.49802 \dots^{10} + 20 \cdot 2.49802 \dots^{9} + 18 \cdot 2.49802 \dots^{8} + 16 \cdot 2.49802 \dots^{7} + 14 \cdot 2.49802 \dots^{6} + 12 \cdot 2.49802 \dots^{5} + 10 \cdot 2.49802 \dots^{4} + 8 \cdot 2.49802 \dots^{3} + 6 \cdot 2.49802 \dots^{2} + 4 \cdot 2.49802 \dots + 1 = 2310601127.77513 \dots

u_{11} = 2.36880 \dots

Δ u_{11} = ∣ 2.36880 \dots - 2.49802 \dots ∣ = 0.12921 \dots

Δ u_{11} = 0.12921 \dots

u_{12} = 2.24587 \dots : Δ u_{12} = 0.12293 \dots

f (u_{11}) = 2 \cdot 2.36880 \dots^{20} + 2 \cdot 2.36880 \dots^{19} + 2 \cdot 2.36880 \dots^{18} + 2 \cdot 2.36880 \dots^{17} + 2 \cdot 2.36880 \dots^{16} + 2 \cdot 2.36880 \dots^{15} + 2 \cdot 2.36880 \dots^{14} + 2 \cdot 2.36880 \dots^{13} + 2 \cdot 2.36880 \dots^{12} + 2 \cdot 2.36880 \dots^{11} + 2 \cdot 2.36880 \dots^{10} + 2 \cdot 2.36880 \dots^{9} + 2 \cdot 2.36880 \dots^{8} + 2 \cdot 2.36880 \dots^{7} + 2 \cdot 2.36880 \dots^{6} + 2 \cdot 2.36880 \dots^{5} + 2 \cdot 2.36880 \dots^{4} + 2 \cdot 2.36880 \dots^{3} + 2 \cdot 2.36880 \dots^{2} + 2.36880 \dots + 1 = 107106520.29046 \dots

f^{^{'}} (u_{11}) = 40 \cdot 2.36880 \dots^{19} + 38 \cdot 2.36880 \dots^{18} + 36 \cdot 2.36880 \dots^{17} + 34 \cdot 2.36880 \dots^{16} + 32 \cdot 2.36880 \dots^{15} + 30 \cdot 2.36880 \dots^{14} + 28 \cdot 2.36880 \dots^{13} + 26 \cdot 2.36880 \dots^{12} + 24 \cdot 2.36880 \dots^{11} + 22 \cdot 2.36880 \dots^{10} + 20 \cdot 2.36880 \dots^{9} + 18 \cdot 2.36880 \dots^{8} + 16 \cdot 2.36880 \dots^{7} + 14 \cdot 2.36880 \dots^{6} + 12 \cdot 2.36880 \dots^{5} + 10 \cdot 2.36880 \dots^{4} + 8 \cdot 2.36880 \dots^{3} + 6 \cdot 2.36880 \dots^{2} + 4 \cdot 2.36880 \dots + 1 = 871274563.57524 \dots

u_{12} = 2.24587 \dots

Δ u_{12} = ∣ 2.24587 \dots - 2.36880 \dots ∣ = 0.12293 \dots

Δ u_{12} = 0.12293 \dots

u_{13} = 2.12888 \dots : Δ u_{13} = 0.11698 \dots

f (u_{12}) = 2 \cdot 2.24587 \dots^{20} + 2 \cdot 2.24587 \dots^{19} + 2 \cdot 2.24587 \dots^{18} + 2 \cdot 2.24587 \dots^{17} + 2 \cdot 2.24587 \dots^{16} + 2 \cdot 2.24587 \dots^{15} + 2 \cdot 2.24587 \dots^{14} + 2 \cdot 2.24587 \dots^{13} + 2 \cdot 2.24587 \dots^{12} + 2 \cdot 2.24587 \dots^{11} + 2 \cdot 2.24587 \dots^{10} + 2 \cdot 2.24587 \dots^{9} + 2 \cdot 2.24587 \dots^{8} + 2 \cdot 2.24587 \dots^{7} + 2 \cdot 2.24587 \dots^{6} + 2 \cdot 2.24587 \dots^{5} + 2 \cdot 2.24587 \dots^{4} + 2 \cdot 2.24587 \dots^{3} + 2 \cdot 2.24587 \dots^{2} + 2.24587 \dots + 1 = 38429268.19821 \dots

f^{^{'}} (u_{12}) = 40 \cdot 2.24587 \dots^{19} + 38 \cdot 2.24587 \dots^{18} + 36 \cdot 2.24587 \dots^{17} + 34 \cdot 2.24587 \dots^{16} + 32 \cdot 2.24587 \dots^{15} + 30 \cdot 2.24587 \dots^{14} + 28 \cdot 2.24587 \dots^{13} + 26 \cdot 2.24587 \dots^{12} + 24 \cdot 2.24587 \dots^{11} + 22 \cdot 2.24587 \dots^{10} + 20 \cdot 2.24587 \dots^{9} + 18 \cdot 2.24587 \dots^{8} + 16 \cdot 2.24587 \dots^{7} + 14 \cdot 2.24587 \dots^{6} + 12 \cdot 2.24587 \dots^{5} + 10 \cdot 2.24587 \dots^{4} + 8 \cdot 2.24587 \dots^{3} + 6 \cdot 2.24587 \dots^{2} + 4 \cdot 2.24587 \dots + 1 = 328486438.92554 \dots

u_{13} = 2.12888 \dots

Δ u_{13} = ∣ 2.12888 \dots - 2.24587 \dots ∣ = 0.11698 \dots

Δ u_{13} = 0.11698 \dots

u_{14} = 2.01751 \dots : Δ u_{14} = 0.11137 \dots

f (u_{13}) = 2 \cdot 2.12888 \dots^{20} + 2 \cdot 2.12888 \dots^{19} + 2 \cdot 2.12888 \dots^{18} + 2 \cdot 2.12888 \dots^{17} + 2 \cdot 2.12888 \dots^{16} + 2 \cdot 2.12888 \dots^{15} + 2 \cdot 2.12888 \dots^{14} + 2 \cdot 2.12888 \dots^{13} + 2 \cdot 2.12888 \dots^{12} + 2 \cdot 2.12888 \dots^{11} + 2 \cdot 2.12888 \dots^{10} + 2 \cdot 2.12888 \dots^{9} + 2 \cdot 2.12888 \dots^{8} + 2 \cdot 2.12888 \dots^{7} + 2 \cdot 2.12888 \dots^{6} + 2 \cdot 2.12888 \dots^{5} + 2 \cdot 2.12888 \dots^{4} + 2 \cdot 2.12888 \dots^{3} + 2 \cdot 2.12888 \dots^{2} + 2.12888 \dots + 1 = 13790835.58464 \dots

f^{^{'}} (u_{13}) = 40 \cdot 2.12888 \dots^{19} + 38 \cdot 2.12888 \dots^{18} + 36 \cdot 2.12888 \dots^{17} + 34 \cdot 2.12888 \dots^{16} + 32 \cdot 2.12888 \dots^{15} + 30 \cdot 2.12888 \dots^{14} + 28 \cdot 2.12888 \dots^{13} + 26 \cdot 2.12888 \dots^{12} + 24 \cdot 2.12888 \dots^{11} + 22 \cdot 2.12888 \dots^{10} + 20 \cdot 2.12888 \dots^{9} + 18 \cdot 2.12888 \dots^{8} + 16 \cdot 2.12888 \dots^{7} + 14 \cdot 2.12888 \dots^{6} + 12 \cdot 2.12888 \dots^{5} + 10 \cdot 2.12888 \dots^{4} + 8 \cdot 2.12888 \dots^{3} + 6 \cdot 2.12888 \dots^{2} + 4 \cdot 2.12888 \dots + 1 = 123820714.41332 \dots

u_{14} = 2.01751 \dots

Δ u_{14} = ∣ 2.01751 \dots - 2.12888 \dots ∣ = 0.11137 \dots

Δ u_{14} = 0.11137 \dots

u_{15} = 1.91142 \dots : Δ u_{15} = 0.10608 \dots

f (u_{14}) = 2 \cdot 2.01751 \dots^{20} + 2 \cdot 2.01751 \dots^{19} + 2 \cdot 2.01751 \dots^{18} + 2 \cdot 2.01751 \dots^{17} + 2 \cdot 2.01751 \dots^{16} + 2 \cdot 2.01751 \dots^{15} + 2 \cdot 2.01751 \dots^{14} + 2 \cdot 2.01751 \dots^{13} + 2 \cdot 2.01751 \dots^{12} + 2 \cdot 2.01751 \dots^{11} + 2 \cdot 2.01751 \dots^{10} + 2 \cdot 2.01751 \dots^{9} + 2 \cdot 2.01751 \dots^{8} + 2 \cdot 2.01751 \dots^{7} + 2 \cdot 2.01751 \dots^{6} + 2 \cdot 2.01751 \dots^{5} + 2 \cdot 2.01751 \dots^{4} + 2 \cdot 2.01751 \dots^{3} + 2 \cdot 2.01751 \dots^{2} + 2.01751 \dots + 1 = 4950229.82773 \dots

f^{^{'}} (u_{14}) = 40 \cdot 2.01751 \dots^{19} + 38 \cdot 2.01751 \dots^{18} + 36 \cdot 2.01751 \dots^{17} + 34 \cdot 2.01751 \dots^{16} + 32 \cdot 2.01751 \dots^{15} + 30 \cdot 2.01751 \dots^{14} + 28 \cdot 2.01751 \dots^{13} + 26 \cdot 2.01751 \dots^{12} + 24 \cdot 2.01751 \dots^{11} + 22 \cdot 2.01751 \dots^{10} + 20 \cdot 2.01751 \dots^{9} + 18 \cdot 2.01751 \dots^{8} + 16 \cdot 2.01751 \dots^{7} + 14 \cdot 2.01751 \dots^{6} + 12 \cdot 2.01751 \dots^{5} + 10 \cdot 2.01751 \dots^{4} + 8 \cdot 2.01751 \dots^{3} + 6 \cdot 2.01751 \dots^{2} + 4 \cdot 2.01751 \dots + 1 = 46661280.69367 \dots

u_{15} = 1.91142 \dots

Δ u_{15} = ∣ 1.91142 \dots - 2.01751 \dots ∣ = 0.10608 \dots

Δ u_{15} = 0.10608 \dots

u_{16} = 1.81030 \dots : Δ u_{16} = 0.10111 \dots

f (u_{15}) = 2 \cdot 1.91142 \dots^{20} + 2 \cdot 1.91142 \dots^{19} + 2 \cdot 1.91142 \dots^{18} + 2 \cdot 1.91142 \dots^{17} + 2 \cdot 1.91142 \dots^{16} + 2 \cdot 1.91142 \dots^{15} + 2 \cdot 1.91142 \dots^{14} + 2 \cdot 1.91142 \dots^{13} + 2 \cdot 1.91142 \dots^{12} + 2 \cdot 1.91142 \dots^{11} + 2 \cdot 1.91142 \dots^{10} + 2 \cdot 1.91142 \dots^{9} + 2 \cdot 1.91142 \dots^{8} + 2 \cdot 1.91142 \dots^{7} + 2 \cdot 1.91142 \dots^{6} + 2 \cdot 1.91142 \dots^{5} + 2 \cdot 1.91142 \dots^{4} + 2 \cdot 1.91142 \dots^{3} + 2 \cdot 1.91142 \dots^{2} + 1.91142 \dots + 1 = 1777460.56654 \dots

f^{^{'}} (u_{15}) = 40 \cdot 1.91142 \dots^{19} + 38 \cdot 1.91142 \dots^{18} + 36 \cdot 1.91142 \dots^{17} + 34 \cdot 1.91142 \dots^{16} + 32 \cdot 1.91142 \dots^{15} + 30 \cdot 1.91142 \dots^{14} + 28 \cdot 1.91142 \dots^{13} + 26 \cdot 1.91142 \dots^{12} + 24 \cdot 1.91142 \dots^{11} + 22 \cdot 1.91142 \dots^{10} + 20 \cdot 1.91142 \dots^{9} + 18 \cdot 1.91142 \dots^{8} + 16 \cdot 1.91142 \dots^{7} + 14 \cdot 1.91142 \dots^{6} + 12 \cdot 1.91142 \dots^{5} + 10 \cdot 1.91142 \dots^{4} + 8 \cdot 1.91142 \dots^{3} + 6 \cdot 1.91142 \dots^{2} + 4 \cdot 1.91142 \dots + 1 = 17578062.54966 \dots

u_{16} = 1.81030 \dots

Δ u_{16} = ∣ 1.81030 \dots - 1.91142 \dots ∣ = 0.10111 \dots

Δ u_{16} = 0.10111 \dots

u_{17} = 1.71383 \dots : Δ u_{17} = 0.09646 \dots

f (u_{16}) = 2 \cdot 1.81030 \dots^{20} + 2 \cdot 1.81030 \dots^{19} + 2 \cdot 1.81030 \dots^{18} + 2 \cdot 1.81030 \dots^{17} + 2 \cdot 1.81030 \dots^{16} + 2 \cdot 1.81030 \dots^{15} + 2 \cdot 1.81030 \dots^{14} + 2 \cdot 1.81030 \dots^{13} + 2 \cdot 1.81030 \dots^{12} + 2 \cdot 1.81030 \dots^{11} + 2 \cdot 1.81030 \dots^{10} + 2 \cdot 1.81030 \dots^{9} + 2 \cdot 1.81030 \dots^{8} + 2 \cdot 1.81030 \dots^{7} + 2 \cdot 1.81030 \dots^{6} + 2 \cdot 1.81030 \dots^{5} + 2 \cdot 1.81030 \dots^{4} + 2 \cdot 1.81030 \dots^{3} + 2 \cdot 1.81030 \dots^{2} + 1.81030 \dots + 1 = 638502.05884 \dots

f^{^{'}} (u_{16}) = 40 \cdot 1.81030 \dots^{19} + 38 \cdot 1.81030 \dots^{18} + 36 \cdot 1.81030 \dots^{17} + 34 \cdot 1.81030 \dots^{16} + 32 \cdot 1.81030 \dots^{15} + 30 \cdot 1.81030 \dots^{14} + 28 \cdot 1.81030 \dots^{13} + 26 \cdot 1.81030 \dots^{12} + 24 \cdot 1.81030 \dots^{11} + 22 \cdot 1.81030 \dots^{10} + 20 \cdot 1.81030 \dots^{9} + 18 \cdot 1.81030 \dots^{8} + 16 \cdot 1.81030 \dots^{7} + 14 \cdot 1.81030 \dots^{6} + 12 \cdot 1.81030 \dots^{5} + 10 \cdot 1.81030 \dots^{4} + 8 \cdot 1.81030 \dots^{3} + 6 \cdot 1.81030 \dots^{2} + 4 \cdot 1.81030 \dots + 1 = 6618867.78758 \dots

u_{17} = 1.71383 \dots

Δ u_{17} = ∣ 1.71383 \dots - 1.81030 \dots ∣ = 0.09646 \dots

Δ u_{17} = 0.09646 \dots

u_{18} = 1.62169 \dots : Δ u_{18} = 0.09214 \dots

f (u_{17}) = 2 \cdot 1.71383 \dots^{20} + 2 \cdot 1.71383 \dots^{19} + 2 \cdot 1.71383 \dots^{18} + 2 \cdot 1.71383 \dots^{17} + 2 \cdot 1.71383 \dots^{16} + 2 \cdot 1.71383 \dots^{15} + 2 \cdot 1.71383 \dots^{14} + 2 \cdot 1.71383 \dots^{13} + 2 \cdot 1.71383 \dots^{12} + 2 \cdot 1.71383 \dots^{11} + 2 \cdot 1.71383 \dots^{10} + 2 \cdot 1.71383 \dots^{9} + 2 \cdot 1.71383 \dots^{8} + 2 \cdot 1.71383 \dots^{7} + 2 \cdot 1.71383 \dots^{6} + 2 \cdot 1.71383 \dots^{5} + 2 \cdot 1.71383 \dots^{4} + 2 \cdot 1.71383 \dots^{3} + 2 \cdot 1.71383 \dots^{2} + 1.71383 \dots + 1 = 229500.02828 \dots

f^{^{'}} (u_{17}) = 40 \cdot 1.71383 \dots^{19} + 38 \cdot 1.71383 \dots^{18} + 36 \cdot 1.71383 \dots^{17} + 34 \cdot 1.71383 \dots^{16} + 32 \cdot 1.71383 \dots^{15} + 30 \cdot 1.71383 \dots^{14} + 28 \cdot 1.71383 \dots^{13} + 26 \cdot 1.71383 \dots^{12} + 24 \cdot 1.71383 \dots^{11} + 22 \cdot 1.71383 \dots^{10} + 20 \cdot 1.71383 \dots^{9} + 18 \cdot 1.71383 \dots^{8} + 16 \cdot 1.71383 \dots^{7} + 14 \cdot 1.71383 \dots^{6} + 12 \cdot 1.71383 \dots^{5} + 10 \cdot 1.71383 \dots^{4} + 8 \cdot 1.71383 \dots^{3} + 6 \cdot 1.71383 \dots^{2} + 4 \cdot 1.71383 \dots + 1 = 2490671.57675 \dots

u_{18} = 1.62169 \dots

Δ u_{18} = ∣ 1.62169 \dots - 1.71383 \dots ∣ = 0.09214 \dots

Δ u_{18} = 0.09214 \dots

u_{19} = 1.53352 \dots : Δ u_{19} = 0.08816 \dots

f (u_{18}) = 2 \cdot 1.62169 \dots^{20} + 2 \cdot 1.62169 \dots^{19} + 2 \cdot 1.62169 \dots^{18} + 2 \cdot 1.62169 \dots^{17} + 2 \cdot 1.62169 \dots^{16} + 2 \cdot 1.62169 \dots^{15} + 2 \cdot 1.62169 \dots^{14} + 2 \cdot 1.62169 \dots^{13} + 2 \cdot 1.62169 \dots^{12} + 2 \cdot 1.62169 \dots^{11} + 2 \cdot 1.62169 \dots^{10} + 2 \cdot 1.62169 \dots^{9} + 2 \cdot 1.62169 \dots^{8} + 2 \cdot 1.62169 \dots^{7} + 2 \cdot 1.62169 \dots^{6} + 2 \cdot 1.62169 \dots^{5} + 2 \cdot 1.62169 \dots^{4} + 2 \cdot 1.62169 \dots^{3} + 2 \cdot 1.62169 \dots^{2} + 1.62169 \dots + 1 = 82559.70843 \dots

f^{^{'}} (u_{18}) = 40 \cdot 1.62169 \dots^{19} + 38 \cdot 1.62169 \dots^{18} + 36 \cdot 1.62169 \dots^{17} + 34 \cdot 1.62169 \dots^{16} + 32 \cdot 1.62169 \dots^{15} + 30 \cdot 1.62169 \dots^{14} + 28 \cdot 1.62169 \dots^{13} + 26 \cdot 1.62169 \dots^{12} + 24 \cdot 1.62169 \dots^{11} + 22 \cdot 1.62169 \dots^{10} + 20 \cdot 1.62169 \dots^{9} + 18 \cdot 1.62169 \dots^{8} + 16 \cdot 1.62169 \dots^{7} + 14 \cdot 1.62169 \dots^{6} + 12 \cdot 1.62169 \dots^{5} + 10 \cdot 1.62169 \dots^{4} + 8 \cdot 1.62169 \dots^{3} + 6 \cdot 1.62169 \dots^{2} + 4 \cdot 1.62169 \dots + 1 = 936373.05744 \dots

u_{19} = 1.53352 \dots

Δ u_{19} = ∣ 1.53352 \dots - 1.62169 \dots ∣ = 0.08816 \dots

Δ u_{19} = 0.08816 \dots

u_{20} = 1.44893 \dots : Δ u_{20} = 0.08458 \dots

f (u_{19}) = 2 \cdot 1.53352 \dots^{20} + 2 \cdot 1.53352 \dots^{19} + 2 \cdot 1.53352 \dots^{18} + 2 \cdot 1.53352 \dots^{17} + 2 \cdot 1.53352 \dots^{16} + 2 \cdot 1.53352 \dots^{15} + 2 \cdot 1.53352 \dots^{14} + 2 \cdot 1.53352 \dots^{13} + 2 \cdot 1.53352 \dots^{12} + 2 \cdot 1.53352 \dots^{11} + 2 \cdot 1.53352 \dots^{10} + 2 \cdot 1.53352 \dots^{9} + 2 \cdot 1.53352 \dots^{8} + 2 \cdot 1.53352 \dots^{7} + 2 \cdot 1.53352 \dots^{6} + 2 \cdot 1.53352 \dots^{5} + 2 \cdot 1.53352 \dots^{4} + 2 \cdot 1.53352 \dots^{3} + 2 \cdot 1.53352 \dots^{2} + 1.53352 \dots + 1 = 29736.26727 \dots

f^{^{'}} (u_{19}) = 40 \cdot 1.53352 \dots^{19} + 38 \cdot 1.53352 \dots^{18} + 36 \cdot 1.53352 \dots^{17} + 34 \cdot 1.53352 \dots^{16} + 32 \cdot 1.53352 \dots^{15} + 30 \cdot 1.53352 \dots^{14} + 28 \cdot 1.53352 \dots^{13} + 26 \cdot 1.53352 \dots^{12} + 24 \cdot 1.53352 \dots^{11} + 22 \cdot 1.53352 \dots^{10} + 20 \cdot 1.53352 \dots^{9} + 18 \cdot 1.53352 \dots^{8} + 16 \cdot 1.53352 \dots^{7} + 14 \cdot 1.53352 \dots^{6} + 12 \cdot 1.53352 \dots^{5} + 10 \cdot 1.53352 \dots^{4} + 8 \cdot 1.53352 \dots^{3} + 6 \cdot 1.53352 \dots^{2} + 4 \cdot 1.53352 \dots + 1 = 351551.64069 \dots

u_{20} = 1.44893 \dots

Δ u_{20} = ∣ 1.44893 \dots - 1.53352 \dots ∣ = 0.08458 \dots

Δ u_{20} = 0.08458 \dots

u_{21} = 1.36746 \dots : Δ u_{21} = 0.08146 \dots

f (u_{20}) = 2 \cdot 1.44893 \dots^{20} + 2 \cdot 1.44893 \dots^{19} + 2 \cdot 1.44893 \dots^{18} + 2 \cdot 1.44893 \dots^{17} + 2 \cdot 1.44893 \dots^{16} + 2 \cdot 1.44893 \dots^{15} + 2 \cdot 1.44893 \dots^{14} + 2 \cdot 1.44893 \dots^{13} + 2 \cdot 1.44893 \dots^{12} + 2 \cdot 1.44893 \dots^{11} + 2 \cdot 1.44893 \dots^{10} + 2 \cdot 1.44893 \dots^{9} + 2 \cdot 1.44893 \dots^{8} + 2 \cdot 1.44893 \dots^{7} + 2 \cdot 1.44893 \dots^{6} + 2 \cdot 1.44893 \dots^{5} + 2 \cdot 1.44893 \dots^{4} + 2 \cdot 1.44893 \dots^{3} + 2 \cdot 1.44893 \dots^{2} + 1.44893 \dots + 1 = 10730.28828 \dots

f^{^{'}} (u_{20}) = 40 \cdot 1.44893 \dots^{19} + 38 \cdot 1.44893 \dots^{18} + 36 \cdot 1.44893 \dots^{17} + 34 \cdot 1.44893 \dots^{16} + 32 \cdot 1.44893 \dots^{15} + 30 \cdot 1.44893 \dots^{14} + 28 \cdot 1.44893 \dots^{13} + 26 \cdot 1.44893 \dots^{12} + 24 \cdot 1.44893 \dots^{11} + 22 \cdot 1.44893 \dots^{10} + 20 \cdot 1.44893 \dots^{9} + 18 \cdot 1.44893 \dots^{8} + 16 \cdot 1.44893 \dots^{7} + 14 \cdot 1.44893 \dots^{6} + 12 \cdot 1.44893 \dots^{5} + 10 \cdot 1.44893 \dots^{4} + 8 \cdot 1.44893 \dots^{3} + 6 \cdot 1.44893 \dots^{2} + 4 \cdot 1.44893 \dots + 1 = 131710.17919 \dots

u_{21} = 1.36746 \dots

Δ u_{21} = ∣ 1.36746 \dots - 1.44893 \dots ∣ = 0.08146 \dots

Δ u_{21} = 0.08146 \dots

u_{22} = 1.28850 \dots : Δ u_{22} = 0.07896 \dots

f (u_{21}) = 2 \cdot 1.36746 \dots^{20} + 2 \cdot 1.36746 \dots^{19} + 2 \cdot 1.36746 \dots^{18} + 2 \cdot 1.36746 \dots^{17} + 2 \cdot 1.36746 \dots^{16} + 2 \cdot 1.36746 \dots^{15} + 2 \cdot 1.36746 \dots^{14} + 2 \cdot 1.36746 \dots^{13} + 2 \cdot 1.36746 \dots^{12} + 2 \cdot 1.36746 \dots^{11} + 2 \cdot 1.36746 \dots^{10} + 2 \cdot 1.36746 \dots^{9} + 2 \cdot 1.36746 \dots^{8} + 2 \cdot 1.36746 \dots^{7} + 2 \cdot 1.36746 \dots^{6} + 2 \cdot 1.36746 \dots^{5} + 2 \cdot 1.36746 \dots^{4} + 2 \cdot 1.36746 \dots^{3} + 2 \cdot 1.36746 \dots^{2} + 1.36746 \dots + 1 = 3883.34198 \dots

f^{^{'}} (u_{21}) = 40 \cdot 1.36746 \dots^{19} + 38 \cdot 1.36746 \dots^{18} + 36 \cdot 1.36746 \dots^{17} + 34 \cdot 1.36746 \dots^{16} + 32 \cdot 1.36746 \dots^{15} + 30 \cdot 1.36746 \dots^{14} + 28 \cdot 1.36746 \dots^{13} + 26 \cdot 1.36746 \dots^{12} + 24 \cdot 1.36746 \dots^{11} + 22 \cdot 1.36746 \dots^{10} + 20 \cdot 1.36746 \dots^{9} + 18 \cdot 1.36746 \dots^{8} + 16 \cdot 1.36746 \dots^{7} + 14 \cdot 1.36746 \dots^{6} + 12 \cdot 1.36746 \dots^{5} + 10 \cdot 1.36746 \dots^{4} + 8 \cdot 1.36746 \dots^{3} + 6 \cdot 1.36746 \dots^{2} + 4 \cdot 1.36746 \dots + 1 = 49180.53699 \dots

u_{22} = 1.28850 \dots

Δ u_{22} = ∣ 1.28850 \dots - 1.36746 \dots ∣ = 0.07896 \dots

Δ u_{22} = 0.07896 \dots

u_{23} = 1.21118 \dots : Δ u_{23} = 0.07732 \dots

f (u_{22}) = 2 \cdot 1.28850 \dots^{20} + 2 \cdot 1.28850 \dots^{19} + 2 \cdot 1.28850 \dots^{18} + 2 \cdot 1.28850 \dots^{17} + 2 \cdot 1.28850 \dots^{16} + 2 \cdot 1.28850 \dots^{15} + 2 \cdot 1.28850 \dots^{14} + 2 \cdot 1.28850 \dots^{13} + 2 \cdot 1.28850 \dots^{12} + 2 \cdot 1.28850 \dots^{11} + 2 \cdot 1.28850 \dots^{10} + 2 \cdot 1.28850 \dots^{9} + 2 \cdot 1.28850 \dots^{8} + 2 \cdot 1.28850 \dots^{7} + 2 \cdot 1.28850 \dots^{6} + 2 \cdot 1.28850 \dots^{5} + 2 \cdot 1.28850 \dots^{4} + 2 \cdot 1.28850 \dots^{3} + 2 \cdot 1.28850 \dots^{2} + 1.28850 \dots + 1 = 1412.13758 \dots

f^{^{'}} (u_{22}) = 40 \cdot 1.28850 \dots^{19} + 38 \cdot 1.28850 \dots^{18} + 36 \cdot 1.28850 \dots^{17} + 34 \cdot 1.28850 \dots^{16} + 32 \cdot 1.28850 \dots^{15} + 30 \cdot 1.28850 \dots^{14} + 28 \cdot 1.28850 \dots^{13} + 26 \cdot 1.28850 \dots^{12} + 24 \cdot 1.28850 \dots^{11} + 22 \cdot 1.28850 \dots^{10} + 20 \cdot 1.28850 \dots^{9} + 18 \cdot 1.28850 \dots^{8} + 16 \cdot 1.28850 \dots^{7} + 14 \cdot 1.28850 \dots^{6} + 12 \cdot 1.28850 \dots^{5} + 10 \cdot 1.28850 \dots^{4} + 8 \cdot 1.28850 \dots^{3} + 6 \cdot 1.28850 \dots^{2} + 4 \cdot 1.28850 \dots + 1 = 18261.62900 \dots

u_{23} = 1.21118 \dots

Δ u_{23} = ∣ 1.21118 \dots - 1.28850 \dots ∣ = 0.07732 \dots

Δ u_{23} = 0.07732 \dots

u_{24} = 1.13409 \dots : Δ u_{24} = 0.07708 \dots

f (u_{23}) = 2 \cdot 1.21118 \dots^{20} + 2 \cdot 1.21118 \dots^{19} + 2 \cdot 1.21118 \dots^{18} + 2 \cdot 1.21118 \dots^{17} + 2 \cdot 1.21118 \dots^{16} + 2 \cdot 1.21118 \dots^{15} + 2 \cdot 1.21118 \dots^{14} + 2 \cdot 1.21118 \dots^{13} + 2 \cdot 1.21118 \dots^{12} + 2 \cdot 1.21118 \dots^{11} + 2 \cdot 1.21118 \dots^{10} + 2 \cdot 1.21118 \dots^{9} + 2 \cdot 1.21118 \dots^{8} + 2 \cdot 1.21118 \dots^{7} + 2 \cdot 1.21118 \dots^{6} + 2 \cdot 1.21118 \dots^{5} + 2 \cdot 1.21118 \dots^{4} + 2 \cdot 1.21118 \dots^{3} + 2 \cdot 1.21118 \dots^{2} + 1.21118 \dots + 1 = 517.69016 \dots

f^{^{'}} (u_{23}) = 40 \cdot 1.21118 \dots^{19} + 38 \cdot 1.21118 \dots^{18} + 36 \cdot 1.21118 \dots^{17} + 34 \cdot 1.21118 \dots^{16} + 32 \cdot 1.21118 \dots^{15} + 30 \cdot 1.21118 \dots^{14} + 28 \cdot 1.21118 \dots^{13} + 26 \cdot 1.21118 \dots^{12} + 24 \cdot 1.21118 \dots^{11} + 22 \cdot 1.21118 \dots^{10} + 20 \cdot 1.21118 \dots^{9} + 18 \cdot 1.21118 \dots^{8} + 16 \cdot 1.21118 \dots^{7} + 14 \cdot 1.21118 \dots^{6} + 12 \cdot 1.21118 \dots^{5} + 10 \cdot 1.21118 \dots^{4} + 8 \cdot 1.21118 \dots^{3} + 6 \cdot 1.21118 \dots^{2} + 4 \cdot 1.21118 \dots + 1 = 6715.60947 \dots

u_{24} = 1.13409 \dots

Δ u_{24} = ∣ 1.13409 \dots - 1.21118 \dots ∣ = 0.07708 \dots

Δ u_{24} = 0.07708 \dots

u_{25} = 1.05480 \dots : Δ u_{25} = 0.07929 \dots

f (u_{24}) = 2 \cdot 1.13409 \dots^{20} + 2 \cdot 1.13409 \dots^{19} + 2 \cdot 1.13409 \dots^{18} + 2 \cdot 1.13409 \dots^{17} + 2 \cdot 1.13409 \dots^{16} + 2 \cdot 1.13409 \dots^{15} + 2 \cdot 1.13409 \dots^{14} + 2 \cdot 1.13409 \dots^{13} + 2 \cdot 1.13409 \dots^{12} + 2 \cdot 1.13409 \dots^{11} + 2 \cdot 1.13409 \dots^{10} + 2 \cdot 1.13409 \dots^{9} + 2 \cdot 1.13409 \dots^{8} + 2 \cdot 1.13409 \dots^{7} + 2 \cdot 1.13409 \dots^{6} + 2 \cdot 1.13409 \dots^{5} + 2 \cdot 1.13409 \dots^{4} + 2 \cdot 1.13409 \dots^{3} + 2 \cdot 1.13409 \dots^{2} + 1.13409 \dots + 1 = 192.47985 \dots

f^{^{'}} (u_{24}) = 40 \cdot 1.13409 \dots^{19} + 38 \cdot 1.13409 \dots^{18} + 36 \cdot 1.13409 \dots^{17} + 34 \cdot 1.13409 \dots^{16} + 32 \cdot 1.13409 \dots^{15} + 30 \cdot 1.13409 \dots^{14} + 28 \cdot 1.13409 \dots^{13} + 26 \cdot 1.13409 \dots^{12} + 24 \cdot 1.13409 \dots^{11} + 22 \cdot 1.13409 \dots^{10} + 20 \cdot 1.13409 \dots^{9} + 18 \cdot 1.13409 \dots^{8} + 16 \cdot 1.13409 \dots^{7} + 14 \cdot 1.13409 \dots^{6} + 12 \cdot 1.13409 \dots^{5} + 10 \cdot 1.13409 \dots^{4} + 8 \cdot 1.13409 \dots^{3} + 6 \cdot 1.13409 \dots^{2} + 4 \cdot 1.13409 \dots + 1 = 2427.51025 \dots

u_{25} = 1.05480 \dots

Δ u_{25} = ∣ 1.05480 \dots - 1.13409 \dots ∣ = 0.07929 \dots

Δ u_{25} = 0.07929 \dots

u_{26} = 0.96859 \dots : Δ u_{26} = 0.08620 \dots

f (u_{25}) = 2 \cdot 1.05480 \dots^{20} + 2 \cdot 1.05480 \dots^{19} + 2 \cdot 1.05480 \dots^{18} + 2 \cdot 1.05480 \dots^{17} + 2 \cdot 1.05480 \dots^{16} + 2 \cdot 1.05480 \dots^{15} + 2 \cdot 1.05480 \dots^{14} + 2 \cdot 1.05480 \dots^{13} + 2 \cdot 1.05480 \dots^{12} + 2 \cdot 1.05480 \dots^{11} + 2 \cdot 1.05480 \dots^{10} + 2 \cdot 1.05480 \dots^{9} + 2 \cdot 1.05480 \dots^{8} + 2 \cdot 1.05480 \dots^{7} + 2 \cdot 1.05480 \dots^{6} + 2 \cdot 1.05480 \dots^{5} + 2 \cdot 1.05480 \dots^{4} + 2 \cdot 1.05480 \dots^{3} + 2 \cdot 1.05480 \dots^{2} + 1.05480 \dots + 1 = 73.34809 \dots

f^{^{'}} (u_{25}) = 40 \cdot 1.05480 \dots^{19} + 38 \cdot 1.05480 \dots^{18} + 36 \cdot 1.05480 \dots^{17} + 34 \cdot 1.05480 \dots^{16} + 32 \cdot 1.05480 \dots^{15} + 30 \cdot 1.05480 \dots^{14} + 28 \cdot 1.05480 \dots^{13} + 26 \cdot 1.05480 \dots^{12} + 24 \cdot 1.05480 \dots^{11} + 22 \cdot 1.05480 \dots^{10} + 20 \cdot 1.05480 \dots^{9} + 18 \cdot 1.05480 \dots^{8} + 16 \cdot 1.05480 \dots^{7} + 14 \cdot 1.05480 \dots^{6} + 12 \cdot 1.05480 \dots^{5} + 10 \cdot 1.05480 \dots^{4} + 8 \cdot 1.05480 \dots^{3} + 6 \cdot 1.05480 \dots^{2} + 4 \cdot 1.05480 \dots + 1 = 850.85072 \dots

u_{26} = 0.96859 \dots

Δ u_{26} = ∣ 0.96859 \dots - 1.05480 \dots ∣ = 0.08620 \dots

Δ u_{26} = 0.08620 \dots

Cannot find solution

The solution is

No Solution for u \in R

The solutions are

u = 0, u = 1

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = 1

sin (x) = 0, sin (x) = 1

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

General solutions for

sin (x) = 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{2} + 2 πn

Graph

Popular Examples

(1+tan^2(x))/(1+sec(x))=sec(x)

-sin^2(x)+2cos(x)-2=0

sin(5x-1)= 4/5

sin^2(x)= 1/36

tan(x)=31

Frequently Asked Questions (FAQ)

What is the general solution for sin^3(x)+sin(x)=2sin^{22}(x) ?
The general solution for sin^3(x)+sin(x)=2sin^{22}(x) is x=2pin,x=pi+2pin,x= pi/2+2pin