Find the result.

Algebra Level 2

( x − 1 ) ( x n + x n − 1 + x n − 2 + ⋯ + 1 ) = ? \large ( x - 1 )( x^n + x^{n - 1} + x^{n - 2} + \cdots + 1) = \ ?

x n + 1 − 1 x^{n + 1} - 1 x n + 1 x^n + 1 x n − 1 x^n - 1 x n + 1 + 1 x^{n + 1} + 1

Sin Ndt by SIN NDT , 14, Vietnam

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2 solutions

Chew-Seong Cheong
Mar 11, 2020

( x − 1 ) ( x n + x n − 1 + x n − 2 + ⋯ + 1 ) = ( x − 1 ) × x n + 1 − 1 x − 1 = x n + 1 − 1 (x-1)(x^n+x^{n-1} + x^{n-2} + \cdots + 1) = (x-1)\times \frac {x^{n+1}-1}{x-1} = \boxed{x^{n+1}-1}

1 Helpful 1 Interesting 1 Brilliant 0 Confused
Ryan S
Apr 3, 2020

Expand, and put the latter term multiplied by x x over the latter term multiplied by 1 1 to subtract. Line up similar terms and cancel away.

x n + 1 + x n + x n − 1 . . . x 1 x^{n+1}+x^n+x^{n-1}\text{ }...\text{ }x^1

− x n + x n − 1 + x n − 2 . . . 1 -\quad\quad\text{ } x^n+x^{n-1}+x^{n-2}\text{ }...\text{ }1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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