Mathmatical Induction Problem (Divisibility)

Kb1jij

1. The problem statement, all variables and given/known data
Use Mathematical Induction to prove that [tex] 12^n + 2(5^{n-1}) [/tex] is divisible by 7 for all [tex] n \in Z^+ [/tex]

2. Relevant equations



3. The attempt at a solution

First, show that it works for n = 1:
[tex] 12^1 + 2 \cdot 5^0 = 14 [/tex] , 14/7 = 2

Next assume:
[tex] 12^k + 2(5^{k-1}) = 7A [/tex]

Then, prove for k + 1:
[tex] 12^{k+1} + 2(5^k) [/tex]

I can't figure out how to prove this. I know that this can be changed to:
[tex] 12 \cdot 12^{k} + 2 \cdot 5 (5^{k-1}) [/tex]
But that doesn't seem to help me much.

I also tried substituting values for 12^k and 5^(k-1) from above:
[tex] 12^k = 7A - 2(5^{k-1}) [/tex]
[tex] 2(5^{k-1}) = 7A - 12^k [/tex]

This doesn't seem too help either, I can reduce it to:
[tex] 189A - (12 \cdot 2(5^{k-1})+5(12^k)) [/tex]

Any suggestions?
Thanks,
Tom

neutrino

Actually, having come till

[tex]12.12^k + 2.5(5^{k-1})[/tex],

the next step should have been

[tex]7.12^k + 5.12^k + 2.5(5^{k-1})[/tex].

Kb1jij

Ah, got it now. Thank you. I don't like these induction problems...