Q15 of 66 Page 12

Prove the following by the principle of mathematical induction:

Let P(n):

For n =1 is true,

P(1):

Since, P(n) is true for n =1

Now, For n = k

P(k): - - - - (1)

We have to show that,

Now,

= using equation (1)

Therefore, P(n) is true for n = k + 1

Hence, P(n) is true for all n ϵ N by PMI

Prove the following by the principle of mathematical induction:

1.3 + 3.5 + 5.7 + … + (2n – 1) (2n + 1)

Prove the following by the principle of mathematical induction:

1.2 + 2.3 + 3.4 + … + n(n + 1)

Prove the following by the principle of mathematical induction:

1² + 3² + 5² + … + (2n – 1)²

Prove the following by the principle of mathematical induction:

a + ar + ar² + … + ar^{n – 1}