Question

If $p\left(n\right):{49}^n+{24}^{n-1}$ is divisible by $p$ for $n\in N,$ then the value of $p$ is
(use principle of mathematical induction)

• A. $25$
• B. $7$
• C. $16$
• D. $4$

Answer 1

The correct option is A $25$

As, $P\left(n\right):{49}^n+{24}^{n-1}$ is divisible by $p$ is true for $n\in N$
it is also true for $n=1,$
$\Rightarrow P\left(1\right):49+1=50$ is divisible by $25.$
Let $P\left(n\right):{49}^n+{24}^{n-1}$ is divisible by $25$ is true for $n,$
for $n=n+1:P(n+1):{49}^{n+1}+{24}^n$
$\Rightarrow P(n+1):24({49}^n+{24}^{n-1})+25\cdot {49}^n$ is divisible by $25$ is true for $n=n+1$
$[\because {49}^n+{24}^{n-1}\text{is divisible by}25]$
So, $P\left(n\right):{49}^n+{24}^{n-1}$ is divisible by $25$ is true.