Prove that 5^(2n - 1) is divisible by 8

Step-by-step explanation:

Prove that 52n − 1 is divisible by 8 for all n ∈ N.

Solution

To prove by induction, we must

1. Show it is true for n = 1

2. Assume true for n and prove it is true for n + 1.

Let P(n) be the statement 52n − 1 is divisible by 8.

When n = 1, 52 − 1 = 24, which is divisible by 8.

This proves P(1).

Suppose that P(n) is true, i.e., 52n − 1 is divisible by 8.

Then 52(n+1) − 1 = 52

· 5

2n − 1 = 52

(52n − 1) + (52 − 1).

Note that 52n −1 is divisible by 8 using P(n) and also 52 −1 = 24 is divisible

by 8. Thus we have proved that P(n + 1) is true.

We conclude that P(n) is true for all n ∈ N.