Thank you for reporting, we will resolve it shortly
Q.
By principle of mathematical induction $3^{2n+2} -8n -9 $ is divisible by
Principle of Mathematical Induction
Solution:
Let $P\left(n\right)$ be the statement given by
$ P\left(n\right) ;3^{2n+2}-8 n- 9$ is divisible by $8$.
For $n = 1$, we get
$P\left(1\right): 3^{2\times1+2} -\left(8\times 1\right)-9 = 64 = 8 \times 8$
which is divisible by $8$.
Let $P\left(k\right)$ be true i.e., $3^{2k+2} -8k-9 = 8\lambda\quad...\left(i\right)$
For $n = k + 1$, we have
$ 3^{2\left(k + 1\right)+2} -8\left(k+1\right) -9$
$= 3^{2k+ 2+2}-8k-8 - 9 = 3^{2k+2} 3^{2} - 8k- 17 $
$= \left(8\lambda+ 8k + 9\right)3^{2} - 8k - 17\quad$[using $\left(i\right)$]
$= \left(8\lambda + 8k+ 9\right)9 - 8k - 17 $
$= 72\lambda + 72k + 81 - 8k - 17$
$= 72\lambda + 64k + 64 = 8\left(9\lambda + 8k + 8\right)$
which is divisible by $8$.
Therefore, $P\left(k + 1\right)$ is true when $P\left(k\right)$ is true.
Hence, from the principle of mathematical induction, the statement is true for all natural numbers $n$.