Use Mathematical induction to prove these divisibility results for all positive integers n:
$\displaystyle 5^n+2(11^n)$ is a multiple of 3
Thanks in advance
$\displaystyle P(1):5^1+2(11^1)=5+22=27$ is a multiple of 3.
let P(m) be true.
$\displaystyle 5^m+2(11^m)=3k$
$\displaystyle 2(11^m)=3k-5^m$
$\displaystyle P(m+1):5^{m+1}+2(11^{m+1})=5^m.5+2(11^m.11)$
= $\displaystyle 5^m.5+(3k-5^m).11$
= $\displaystyle 5^m.5+3k.11-5^m.11$
= $\displaystyle 33k-6(5^m)$, which is divisible by 6.
Thus, P(m+1) is true whenever P(m) is true.
Hence, by the principle of mathematical induction, $\displaystyle 5^n+2(11^n)$ is a multiple of 3 for all positive integers n.