Question

$5^{2n+2}-24n-25$ is divisible by 576 for all n $ϵ$ N.

Answer 1

Let P(n) : $5^{2n+2}-24n-25$ is divisible by 576

For n = 1

$5^4-24-25$

$=625-49=576$

Which is divisible by 576

Let P(n) is true for n = k, so

$5^{2k+2}-24k-25$ is divisible by 576

$5^{2k+2}-24k-25=576\lambda$ ........(1)

We have to show that,

$5^{2k+4}-24(k+1)-25$ is divisible by 576

$5^{(2k+2)+2}-24(k+1)-25=576\mu$

Now,

$5^{(2k+2)+2}-24(k+1)-25$

$=5^{(2k+2)}{.5}^2-24k-24-25$

$=(576\lambda +24k+25)25-24k-49$

[Using equation (1)]

$=25.576\lambda +600k+625-24k-49$

$=25.576\lambda +576k+576$

$=576(25\lambda +k+1)$

$=576\mu$

$\Rightarrow$ P(n) is true for n = k + 1

$\Rightarrow$ P(n) is true for all n $ϵ$ N by PMI.