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

Explanation

`P(n) : 3^(2n+2) - 8n -9 = 8x, x in N`
For `n = 1`
` 3^(2 xx 1+2)-8 xx 1 - 9 = 64`
So, `P(1)` is true,
Let `P(k)` be true,
i.e, `3^(2k+2)-8k-9 = 8m`, where `m in N"……"(1)`
We have to prove that `P(k+1)` is true.
i.e, `3^(2(k+1)+2)-8(k+1) - 9 = 8p`, where `p in N`
Now, `= 9 xx 3^(2k+2)-8k - 17`
`= 9 xx (8m+8k+9) - 8k - 17` [Using (1)]
`= 8(8m+8k)+81-8k-17`
`= 9(8m+8k)-8k+64`
`= 8[9(m+k)-k+8]`
Thus, `P(k+1)` is true whenever `P(k)` is true.
Hence, by the principle of mathematical induction, statement `P(n)` is true for all natural numbers.