Q.Prove the following by using the principle of mathematical induction for all $n \in N$: $3^{2n+2} - 8n - 9$ is divisible by 8.

Let $P(n)$ be the statement: $3^{2n+2}-8n-9$ is divisible by $8$.

Base case: For $n=1$,

divisible by $8$. So $P(1)$ is true.

Inductive step: Assume $P(k)$ is true for some $k\ge1$, i.e. there is an integer $m$ with

We must show $3^{2(k+1)+2}-8(k+1)-9=3^{2k+4}-8k-17$ is divisible by $8$.

Write $3^{2k+4}=9\cdot3^{2k+2}$, and use the hypothesis $3^{2k+2}=8m+8k+9$:

Expand:

Factor out $8$: …

Divisibility induction. Isolate $3^{2k+2}=8m+8k+9$ from the induction hypothesis, substitute into $3^{2k+4}=9\cdot3^{2k+2 …

• Errors relating $3^{2(k+1)+2}$ to $3^{2k+2}$: the exponent increases by $2$, i.e. multiplication by $3^2=9$, not by $3$.
• Arithmetic slip expanding $9(8m+8k+9)=72m+72k+81$; a common error is mis-multiplying $9\times9=81$ or $9\times8k=72k$. …