Embibe Experts Solutions for Chapter: Mathematical Induction, Exercise 1: Exercise 1

A student was asked to prove a statement by induction. He proved

(i) $P\left(5\right)$ is true and

(ii) Truth of $P\left(n\right)\Rightarrow$truth of $p\left(n+1\right), n\in N$

On the basis of this, he could conclude that $p\left(n\right)$ is true for

For every natural number $n$

If $n\in N$, then $x^{2n-1}+y^{2n-1}$ is divisible by-

If $n\in N,$ then ${11}^{n+2}+{12}^{2n+1}$ is divisible by

If $n\in N$, then $3^{4n+2}+5^{2n+1}$ is a multiple of-

For every positive integer $n,\frac{n^7}{7}+\frac{n^5}{5}+\frac{2n^3}{3}-\frac{n}{105}$ is-

For all positive integral values of $n, 3n^{2n}-2n+1$ is divisible by-

If $p$ is a prime number then $n^p-n$ is divisible by $p$ when $n$ is a