Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.n = 54n

What will be an ideal response?

First we show that the statement is true when n = 1.
For n = 1, we get 5⁴ = 5^(4?1) = 5⁴
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)} = 5^4k^{(k + 1)}
^{(k + (k + 1))} = 5^4k
^{(2k + 1)} = 5^{(4k + 4k + 4))}
5^{(4(2k + 1))} = 5^{(4(2k + 1))}
Condition II is satisfied. As a result, the statement is true for all natural numbers n.