Q.Using the principle of mathematical induction, prove that $7^{n} - 3^{n}$ is divisible by $4$ for every natural number $n$.

This is a divisibility induction: the inductive step rewrites $7^{k+1}-3^{k+1}$ so that the assumed multiple of 4, $7^k-3^k$, appears explicitly inside it.

Basis step ($n=1$): $7^1-3^1 = 7-3=4$, which is divisible by $4$. So $P(1)$ is true.

Inductive step: assume $P(k)$ is true, i.e. $7^k-3^k = 4m$ for some integer $m$.

Consider $7^{k+1}-3^{k+1}$. Rewrite it so that $7^k-3^k$ appears explicitly, by adding and subtracting $7\cdot3^k$:

Since $7m+3^k$ is an integer, $7^{k+1}-3^{k+1}$ is a multiple of $4$, so $P(k+1)$ is true whenever $P(k)$ is.

By induction, $7^n-3^n$ is divisible by $4$ for every natural number $n$. …

Trying to prove divisibility by simply checking a handful of values of $n$ and stopping there — checking specific cases (as in the verification step above) is a useful confidence check, but it is never a substitute for th …