Use the method of mathematical induction to prove that $\displaystyle 4^n+14$ is a multiple of $\displaystyle 6$ for $\displaystyle n \geq 1$.
Thanks
Let's see if this holds water.
$\displaystyle 4^{1}+14=18$, true.
Assume 6 is a factor of $\displaystyle 4^{k}+14$. The (k+1)st term is
$\displaystyle 4^{k+1}+14=4\cdot{4^{k}}+14$
=$\displaystyle 4\cdot{4^{k}}+56-42=4(4^{k}+14)-42$
By the induction hypothesis, 6 is a factor of $\displaystyle 4^{k}+14$ and
6 is a factor of 42, so 6 is a factor of the (k+1)st term and $\displaystyle P_{k+1}
$ is true. And QED.