let p(n)=n-3^(2n+2)-8n-9 =8m (let m be any value)
checking for p(1),
lhs: p(1)= 3^(2+2) -8-9=81-17=64=8x8=8m =rhs
therefore p(1)is true
assume p(k)is true .
=>3^(2k+2)-8k-9=8m
\[=>3^{(2k)} = \frac{8m+8k+9}{3^{2}}\]-----(1)
to prove :p(k+1)is true
substitute k+1 in n in the given eq
\[=>3^{2k+4}-8(k+1)-9= 3^{2k}.3^{4}-8k-17\]substitute (1) instead of 3^(2k)
=>72m+64k+64=
=>8(9m+8k+8)
therefore p(k+1) is true
by principal of mathematical induction , p(n) is true for all n belonging to natural numbers