Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
n = 32n
What will be an ideal response?
First we show that the statement is true when n = 1.
For n = 1, we get 32 = 3(2?1) = 32
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) = 32k
(k + 1)
(k + (k + 1)) = 32k
(2k + 1) = 3(2k + 2k + 2))
3(2(2k + 1)) = 3(2(2k + 1))
Condition II is satisfied. As a result, the statement is true for all natural numbers n.
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What will be an ideal response?