Question

Using PMI, prove that $3^{2n+2}-8n-9$ is divisible by $64$.

Answer 1

Let $p\left(x\right)=3^{2n+2}-8x-9$ is divisible by $64$ …..(1)

When put $n=1$,

$p\left(1\right)=3^4-8-9=64$ which is divisible by $64$

Let $n=k$ and we get

$p\left(k\right)=3^{2k+2}-8k-9$ is divisible by $64$

$3^{2k+2}-8k-9=64m$ where $m\in N$ …..(2)

Now we shall prove that $p(k+1)$ is also true

$p(k+1)=3^{2(k+1)+2}-8(k+1)-9$ is divisible by $64$.

Now,

$p(k+1)=3^{2(k+1)+2}-8(k+1)-9=3^2{.3}^{2k+2}-8k-17$

$={9.3}^{2k+2}-8k-17$

$=9(64m+8k+9)-8k-17$

$=9.64m+72k+81-8k-17$

$=9.64m+64k+64$

$=64(9m+k+1),$ Which is divisibility by $64$

Thus $p(k+1)$is true whenever $p\left(k\right)$ is true.

Hence, by principal mathematical induction,

$p\left(x\right)$ is true for all natural number $p\left(x\right)=3^{2n+2}-8x-9$ is divisible by 64 $n\in N$