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 .

Condition I:

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

Left hand side .

Right hand side .

The statement is true for .

Condition 1 of the Principle of Mathematical Induction holds.

Step 2:

Condition II : 

Assume that holds for some ,

and determine whether the formula then holds for .

Assume that, for some ........equation(1).

Now, show that,

for some .

Left hand side from equation(1)

Thus, Condition II also holds. 

The statement is true for all natural numbers.

Solution:

The statement is true for all natural numbers.