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

Prove the following by using the principle of mathematical induction f

1.	Prove the following by using the principle of mathematical induction for all `n in N`:`3^(2n+2)-8n-9`is divisible by 8.
Answer» Given expression is. `3^(2n+2)-8n-9` When `n = 1`, given expression is, `3^4-8-9 = 81-17 = 64` So, for `n=1`, given expession is divisible by `8`. Let for any `k in N`, given expression is divisible by `8`. Then, `3^(2k+2)-8k-9 = 8c->(1)`, where `c` is a natural number. Now, we have to prove, for `n = k+1`, given expression is divisible by `8`. For `n = k+1`, given expression is, `3^(2(k+1)+2)-8(k+1)-9` `= 3^2*3^(2k+2) - 8k -17` `=9(3^(2k+2)-8k-9) +72k+81-8k-17` `=9(3^(2k+2)-8k-9)+64k+64` From (1), `=9(8c)+8k(8k+8)` `=8(9c+8k+8)`, which is clearly divisible by `8`. Thus, given expression will be divisible by `8`.