Question

If n is a positive integer, then $5^{2n+2}-24n-25$ is divisible by

• A. $574$
• B. $575$
• C. $674$
• D. $576$

Answer 1

Answer: (D)

Let $P(n)$ be the statement given by
$P(n):5^{2n+2}-24n-25$ is divisible by $576.$
For $n=1,$
$P(1):5^{2+2}-24-25=625-49=576,$ which is divisible by $576.$
$\therefore P(1)$ is true.
Let $P(k)$ be true,
i.e. $P(k):5^{2k+2}-24k-25$ is divisible by $576.$
$\Rightarrow 5^{2k+2}-24k-25=576\lambda ...\left(i\right)$
We have to show that $P\left(k+1\right)$ is true,
i.e. $5^{2k+4}-24k-49$ is divisible by $576$
Now, $5^{2k+4}-24k-49$
$=\left(576\lambda +24k+25\right)\cdot 25-24k-49$ [from (i)] $=576.25\lambda +600k+625-24k-49$
$=576\left\{25\lambda +k+1\right\}$, which is divisible by $576$.
$\therefore P\left(k+1\right)$ is true whenever $P\left(k\right)$ is true.
So, $P\left(n\right)$ is true for all $n\in N.$