First slide

Introduction to P.M.I

Question

For $n \in N, x^{n + 1} + (x + 1)^{2 n - 1}$ is divisible by

Moderate

A

$x$
B

$x + 1$
C

$x^{2} + x + 1$
D

$x^{2} - x + 1$

Solution

For $n = 1,$ we have
$x^{n + 1} + (x + 1)^{2 n - 1}$
$= x^{2} + (x + 1) = x^{2} + x + 1,$ which is divisible by $x^{2} + x + 1 .$
For $n = 2,$ we have
$\begin{aligned} x^{n + 1} + (x + 1)^{2 n - 1} \\ = x^{3} + (x + 1)^{3} \end{aligned}$
$= (2 x + 1) (x^{2} + x + 1),$ which is divisible by $x^{2} + x + 1 .$
Hence, option (c) is true.

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