Question

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

Answer 1

Step 1:

The statement is is divisible by .

Condition I:

First show that, the above statement is true, when .

is divisible by .

The statement is true for .

Condition I of the Principle of Mathematical Induction holds.

Step 2:

Condition II :

Assume that   is divisible by .holds for some , and determine whether the formula then holds for .

Assume that, is divisible by for some .

Now need show that, is divisible by .

is divisible by and is divisible by .

Therefore, is divisible by .

Thus, Condition II also holds.

The statement is true for all natural numbers.

Solution:

The statement is true for all natural numbers.