For

n = 0

3^{4} + 7 = 88

which is divisible by 11.

If $3^{(3 k + 4)} + 7^{(2 k + 1)} = 11 a$ where $a$ is an integer, then

$3^{(3 (k + 1) + 4)} + 7^{(2 (k + 1) + 1)} = 27 \times 3^{(3 k + 4)} + 49 \times 7^{(2 k + 1)} = 27 \times 11 a + 22 \times 7^{(2 k + 1)}$

and this is divisible by 11.

So, by the axiom of induction, $3^{(3 n + 4)} + 7^{(2 n + 1)}$ is divisible by 11 for all positive integer values of $n$ .