# Math Help - Maths Induction

1. ## Maths Induction

Use Mathematical induction to prove these divisibility results for all positive integers n:

$5^n+2(11^n)$ is a multiple of 3

2. Originally Posted by deltaxray
Use Mathematical induction to prove these divisibility results for all positive integers n:

$5^n+2(11^n)$ is a multiple of 3

$P(1):5^1+2(11^1)=5+22=27$ is a multiple of 3.

let P(m) be true.

$5^m+2(11^m)=3k$

$2(11^m)=3k-5^m$

$P(m+1):5^{m+1}+2(11^{m+1})=5^m.5+2(11^m.11)$

= $5^m.5+(3k-5^m).11$

= $5^m.5+3k.11-5^m.11$

= $33k-6(5^m)$, which is divisible by 6.

Thus, P(m+1) is true whenever P(m) is true.

Hence, by the principle of mathematical induction, $5^n+2(11^n)$ is a multiple of 3 for all positive integers n.