Q.Using the principle of mathematical induction, prove that $7^n - 3^n$ is divisible by 4 for all $n \in N$.

Prove the base case $n=1$, then show that if the statement holds for $n=k$ it must hold for $n=k+1$, using the principle of mathematical induction.

Statement

$P(n) : 7^n - 3^n \text{ is divisible by } 4$

Base case: $P(1)$

$7^1 - 3^1 = 7-3 = 4$

$4$ is divisible by $4$, so $P(1)$ is true.

Inductive step

Assume $P(k)$ is true for some $k\in N$, i.e.

$7^k - 3^k = 4m \quad \text{for some integer } m$

We show $P(k+1)$ is true, i.e. $7^{k+1}-3^{k+1}$ is divisible by 4.

$7^{k+1}-3^{k+1} = 7\cdot 7^k - 3\cdot 3^k$

Write $3\cdot 3^k = 7\cdot 3^k - 4\cdot 3^k$:

$7^{k+1}-3^{k+1} = 7\cdot 7^k - 7\cdot 3^k + 4\cdot 3^k = 7(7^k-3^k) + 4\cdot 3^k$

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