Prove by induction that $3^n +7^n −2$ is divisible by $8$ for all positive integers $n$.
So far I have the base case completed, and believe I am close to completing the proof itself.
Base case:$(n=1)$
$3^1 + 7^1 – 2 = 8/8 = 1 $
Inductive Hypothesis: Assume that $3^n +7^n −2$ is divisible by 8 for all positive integers n.
Induction step $(n+1)$ case:
$$ 3^{n+1} + 7^{n+1} – 2 $$
$$3(3^{n}) + 7(7^{n}) – 2$$
$$3^n + 7^n = 8x $$
-It seems to me that this could be the end of the proof because whatever the answer is would be a multiple of 8: but I am unsure, any help is appreciated.
Best Answer
It holds for $n=1,2$.
If it holds for $1,2,\dots,n$, then
\begin{align} &3^{n+1}+7^{n+1}-2\\ &=3^2\cdot3^{n-1}+7^2\cdot7^{n-1}-2\\ &=(8+1)\cdot3^{n-1}+(48+1)\cdot7^{n-1}-2\\ &=8\cdot(3^{n-1}+6\cdot7^{n-1})+3^{n-1}+7^{n-1}-2\\ \end{align} Therefore it also holds for $n+1$.
So it holds for all $n\in \mathbb N$.