X^0 + x^1 +x^2 + x^3 + .........x^n = 1 find x

2
by sandossh

2015-04-14T14:18:57+05:30

use zero product property and solve
2015-04-14T18:28:58+05:30

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X° + x¹ + x² + x³ + x⁴ + ..... + x^n = 1

this is a Geometric series with starting term x° = 1, and ratio = x.  there are n terms.  The sum is :

S = 1 (x^n+1  - 1) / (x - 1)
= 1

=>  x^n+1 - 1 = x - 1
=>  x^n+1 = x
=> x * x^n = x          => x ( x^n -1 ) = 0

=> x = 0 or,    x^n = 1
So the solutions are  :    x = 0  or the  n th roots of 1, but x ≠ 1.

For example if  n =3,  then x = 0 or the cube roots of 1,  but x ≠ 1.
============

1 + x + x² + x³ + ... + x^n = 1
x ( 1 + x + x² + x³... + x^n-1 ) = 0    ----  (1)
= >    x = 0  or  (x^n -1 ) / (x -1) = 0
=>    x = 0  or  x^n = 1          and  x ≠ 1
=>   x = n th roots of 1.

for example  x = ω or ω² for n = 3.
as 1 + ω + ω² = 0, the above equation (1) is satisfied.
so then roots will be  0 , ω or ω².