First slide

Introduction to P.M.I

Question

$x (x^{n - 1} - {nα}^{n - 1}) + α^{n} (n - 1)$ is divisible by $(x - α)^{2}$ for

Moderate

A

n>1
B

n>2
C

all $n \in N$
D

None of these

Solution

Let $P (n) = x (x^{n - 1} - {nα}^{n - 1}) + α^{n} (n - 1) = (x - α)^{2} g (x)$
$P (1) \equiv 0$ 0istrue.
$\begin{matrix} Let x (x^{k - 1} - {kα}^{k - 1}) + α^{k} (k - 1) = (x - α)^{2} g (x) \\ P (k + 1) \equiv x [x^{k} - (k + 1) α^{k}] + α^{k + 1} (k) \\ \equiv (x - α)^{2} [xg (x) + {kα}^{k - 1}] \end{matrix}$
So, holds for all $n \in N$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App