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

Let `P(n) =3^(2n+2) -8n-9`
for n=1
`p(1) =3^(4)-8(1) -9 =81 -17 =64 =8 (8)`
Which is divisible by 8
`rArrP (n)` be true for n=1
Let P (n) be true for n=K.
`:.P(k) : 3^(2K+2) -8k-9=8lambda (" say ")`
`" Where " lambda in I`
for n=k+1
`P(k+1):3^(2(k+1)+2) -8(k+1)-9`
`=3^(2).3^(2k+2) -8k-8-9`
`=9[8lambda+8k+9]-8k-17`
`=9.8lambda+72k+81-8K-17`
`=9.8lambda+64lambda+64`
`=8[9lambda+8k+8]`
which is divisible by 8
`rArr` P (n) is also true for n=K+1
Hence from the principle of mathematical induction p(n) is true for all natural numbes n.