Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
is divisible by 2
What will be an ideal response?
First, we show that the statement is true when n = 1.
For
This is a true statement and Condition I is satisfied.
Next, we assume the statement holds for some k. That is,
is divisible by 2 is true for some positive integer k.
We need to show that the statement holds for . That is, we need to show that
is divisible by 2.
So we assume is divisible by 2 and look at the expression for n = k + 1.
Since is divisible by 2, then
for some integer m. Hence,
Condition II is satisfied. As a result, the statement is true for all natural numbers n.
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