# Aussie Introduction

#### Damo

##### New member
G'day everyone!
My name is Damo, I'm a secondary school maths teacher in Far North Queensland in Australia. Stumbled across this site looking how to use mathematical induction to prove 34n-1 is divisible by 80. Now I hope I can learn and contribute more.

#### MarkFL

Staff member
Welcome to MHB, Damo!

I hope you find your time here enjoyable and productive!

#### soroban

##### Well-known member
G'day, Damo!

Prove by mathematical induction that $$\:3^{4n}-1$$ is divisible by 80.

Verify $$S(1)\!:\;3^4-1 \:=\:80\;\text{ . . . True!}$$

Assume $$S(k)\!:\;3^{4k}- 1 \;=\;80a\,\text{ for some integer }a.$$

Add $$80\!\cdot\!3^{4k}$$ to both sides.

$\qquad 3^{4k}-1 + 80\!\cdot\!3^{4k} \;=\;80a + 80\!\cdot\!3^{4k}$

$\qquad 3^{4k} - 1 + (3^4-1)3^{4k} \;=\;80(a + 3^{4k})$

$\qquad 3^{4k} - 1 + 3^{4k+4} - 3^{4k} \;=\;80(a+3^{4k})$

$\qquad 3^{4(k+1)} - 1 \;=\;80b\;\text{ for some integer }b$

And we have proved $S(k+1).$
The inductive proof is complete.