Mathmatical Induction Problem (Divisibility)

[Kb1jij]

Homework Statement

Use Mathematical Induction to prove that 12^n + 2(5^{n-1}) is divisible by 7 for all n \in Z^+

Homework Equations

The Attempt at a Solution

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

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

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

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

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

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

Any suggestions?
Thanks,
Tom

 

[neutrino]

Actually, having come till

12.12^k + 2.5(5^{k-1}),

the next step should have been

7.12^k + 5.12^k + 2.5(5^{k-1}).

 

[Kb1jij]

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