Question

Use mathematical induction to prove the truth of each of the following assertions for all n ≥1.

5^2n – 2^5n is divisible by 7

If n = 1, then 5^2(1) - 2^5(1) = -7, which is divisible by 7. For the inductive case, assume k ≥ 1, and the result is true for n = k; that is 7 | (5^2k + 2^5k). Use the assumption to prove n = k + 1, in other words, 5^(2(k + 1)) - 2^(5(k + 1)) is divisible by 7. Now,

5^(2(k + 1)) - 2^(5(k + 1))
= 5^(2k + 2) - 2(5k + 5)
= 5^(2k) · 5^2 - 2^(5k) · 2^5
= 25 · 5^(2k) - 32 · 2^(5k)
= IDK what to do from here. . .

Any suggestions? Thank you again!

Answer 1

Let's continue:

25 · 5^(2k) - 32 · 2^(5k)
=25*(5^(2k)-2^(5k) -7*2^(5k)

Now ask yourself:
A. Is (5^(2k)-2^(5k) divisible by 7, and why?
B. Is -7*2^(5k) divisible by 7, and why?