Proof by Induction - Divisibility Proofs

odolwa99

1. The problem statement, all variables and given/known data

Q. Prove by induction that... (please see attachment).

2. Relevant equations



3. The attempt at a solution

The end result should be divisible by 6, but hasn't worked out for me. Can someone help me spot where I've gone wrong? Thank you.

Attached Thumbnails

 

Mark44

You are using implication symbols (==>) incorrectly. Most of the places you have them, you should be using "=".

As to your question,
7*7^k + 4*4^k + 1
= 7*7^k + 7*4^k + 7 - 3*4^k - 6.

Can you show that the expression above has a factor of 6 in it?

odolwa99

May I ask how you solved for...
7*7k + 7*4k + 7 - 3*4k - 6
from
7*7k + 4*4k + 1?

Mark44

I didn't "solve" for it; I just rewrote the second expression in a different way. You should convince yourself that the two expressions are equal.