Prove the following by the principle of mathematical induction:`\ 3^(2n+2)-8n-9` is divisible 8 for all `n in Ndot`

Explanation

Let the given statement be `{P}(n)`, i.e.,


`{P}(n): 3^{2 n+2}-8 n-9` is divisible by `8`


It can be observed that `{P}(n)` is true for `n=1` since `3^{2 times 1+2}-8 times 1-9=64`, which is divisible by `8` .


Let `P(k)` be true for some positive integer `k`, i.e.,


`3^{2 k+2}-8 k-9` is divisible by `8`


`therefore 3^{2 k+2}-8 k-9=8 m ;` where `m in {N} ldots` (1)


We shall now prove that `{P}(k+1)` is true whenever `{P}(k)` is true.


Consider


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