Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.n = 52n

Question

What will be an ideal response?

Answer 1

First we show that the statement is true when n = 1.
 For n = 1, we get 5² = 5^(2?1) = 5²
This is a true statement and Condition I is satisfied.

Next, we assume the statement holds for some k. That is,
  is true for some positive integer k.
We need to show that the statement holds for k + 1. That is, we need to show that
 
So we assume that  is true and multiply the next term,  to both sides of the equation.
 ^{k(k + 1)} = 5^{2k(k + 1)
 (k + (k + 1))} = 5^{2k
 (2k + 1)} = 5^{(2k + 2k + 2))}
 5^{(2(2k + 1))} = 5^{(2(2k + 1))}
Condition II is satisfied. As a result, the statement is true for all natural numbers n.