Could someone please check this induction proof, unsure if this is the correct way to go about it.

Prove by induction that $\displaystyle 9 \times 4^n + 5 \times 11^n$ is divisible by 7 for all non negative integers $\displaystyle n$.

Initial step, $\displaystyle n=0$

$\displaystyle 9 \times 4^0 + 5 \times 11^0 = 14$, therefore true for $\displaystyle n=0$.

Induction step. Assume for $\displaystyle n=k$, therefore $\displaystyle 9 \times 4^k + 5 \times 11^k = 7a$ for some $\displaystyle a$.

Therefore for $\displaystyle n=k+1$ we have:

$\displaystyle 9 \times 4^{k+1} + 5 \times 11^{k+1}$

$\displaystyle 9(4^k) + 9(4) + 5(11^k) + 5(9)$

What, in the holy name of the Great Pumpkin, did you do here??!: $\displaystyle 9\cdot 4^{k+1}=9\cdot 4^k\cdot 4=36\cdot 4^k$, and the same with the other term, so you need to correct this QUICKLY and think over your proof again. Tonio
$\displaystyle 9(4^k) + 5(11^k) + 9(4) + 5(9)$

$\displaystyle 7a + 91$

$\displaystyle 7(a + 13)$.

Therefore true for all $\displaystyle n$.