Prove by induction that for all $n$, $8$ is a factor of $7^{2n+1} +1$ I want to prove by induction that for all $n$, 8 is a factor of $$7^{2n+1}+1$$
I have proved it true for the base case and assumed it true for $n=k$, but when I cannot figure out when to go towards the end of proving it true for $n=k+1$ assuming it is true for $n=k$.
I let $$7^{2k+1}+1 = 8m$$
Then I work with $$7^{2(k+1)+1}+1$$ to get eventually $$7^2(8m)$$
I am not sure if this is correct or if it is I am not sure how to prove that this too is dividable by $8$.
I would appreciate any help.
Thanks

Your approach was right until you considered $7^{2k+1}+1=8m$
$49 \times 7^{2k+1} + 1 = 49 \times(8m-1) +1 = 8 \times 49m -48$ which is divisible by 8.
