Welcome to our community

Be a part of something great, join today!

Aussie Introduction

Damo

New member
Feb 17, 2015
4
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

Administrator
Staff member
Feb 24, 2012
13,734
Welcome to MHB, Damo! :D

I hope you find your time here enjoyable and productive! (Yes)
 

soroban

Well-known member
Feb 2, 2012
409
G'day, Damo!

Prove by mathematical induction that [tex]\:3^{4n}-1[/tex] is divisible by 80.

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

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


Add [tex]80\!\cdot\!3^{4k}[/tex] 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.