Question

for any positive integer n, prove that$;7^n-3^n$ divisible by 4.

Answer 1

Let $P\left(n\right):7^n-3^n=4d$, where $d\in N$

For $n=1$

LHS = $7-3$

$=4\times 1$

$=RHS$

$P\left(n\right)$ is true for $n=1$.

Assume $P\left(k\right)$ is true.

$P\left(k\right):7^k-3^k=4m$, where $m\in N$

We wiil prove that $P(k+1)$ is true.

$LHS=7^{k+1}-3^{k+1}$

$=7^k.7-3^k.3$

$=7.(4m+3^k)-{3.3}^k$

$=7\times 4m+7\times 3^k-{3.3}^k$

$=7\times 4m+3^k\left(4\right)$

$=4(7m+3^k)$

$=4r$ where $r=7m+3^k$ is a natural number.

Therefore, $P(k+1)$ is true whenever $P\left(k\right)$ is true.

Therefore, by the principle of mathematical induction, $P\left(n\right)$ is true for $n$, where $n$ is a natural number.