Question

$3^{2n+2}-8n-9\text{is divisible by 8.}$

Answer 1

Let Pn) $3^{2n+2}-8n-9\text{is divisible by 8.}$
For n = 1
$P\left(1\right)=3^{2\times 1+2}-8\times 1-9\text{is divisible by 8}\Rightarrow 3^4-8-9\text{is divisible by 8}\Rightarrow 64\text{is divisible by 8}\therefore P\left(1\right)\text{is true}\text{Let P (n) be true for n = k}\therefore P\left(k\right)=3^{2k+2}-8k-9\text{is divisible by 8}$
$\Rightarrow 3^{2k+2}-8k-9=8\lambda \Rightarrow 3^{2k+2}=8\lambda +8k+9\text{For}n=k+1P(k+1)=3^{(k+1)+2}-8(k+1)-9\text{is divisible by 8}=3^{2k+2}.3^2-8k-8-9=(8\lambda +8k+9)9-8k-17=72\lambda +72k+81-8k-17=72\lambda +64k+64=8(9\lambda +8k+8)\Rightarrow 3^{2(k+1)+2}-8(k+1)-9\text{is divisible by 8}\therefore P(k+1)\text{is true}\Rightarrow P(k+1)\text{is true}$