Proving Inequality using Induction.

Question

I am trying to prove the following statement:

For every nonnegative integer $n$, $1+6n \le 7^n$.

I did the base case where $n=0$ but am having trouble manipulating the inductive step. So far I have $7^n+6$. I need that $6$ to be a $7$ so I can have $7^{n+1}$. I am not sure how to make that happen though.

Thanks for the help.

Answer 1

$7^n+6\leq7^n+7\leq7^n+7^n\leq2\cdot7^n\leq7\cdot7^n=7^{n+1}$ if $n\geq1$