Yuo should note that and each term is divisible by 3.This is as far as I have gotten. Any recommendations would be much appreciated.
Proposition. For each natural number n, 3 divides 5^n-2^n.
Proof. We will use the Principle of Mathematical Induction. We let P(n) be “3 divides 5^n-2^n.”
For the basis step, we must prove that P(1) is true. We note that 5^1-2^1=3=3(1) and 1 is an integer. Therefore, we know that 3 divides 5^1-2^1 as 3(1) = 5^1-2^1, and consequently we can conclude that P(1) is true.
For the inductive step, we prove that for all natural numbers k, if P(k) is true, then P(k+1) is true. So let k be a natural number and assume that P(k) is true. That is assume that,
Therefore, we know that there exists an integer m such that
(5^k-2^k )=3m or 5^k=3m+2^k. (1)
In order to prove that P(k+1) is true, we must show that 3|5^(k+1)-2^(k+1). Multiplying both sides of equation (1) by 5 gives us,