**Question 1: **Using principle of mathematical induction prove $1^{3}+2^{3}+3^{3}+ ........... + n^{3}$ =

$\frac{n^{2}(n+1)^{2}}{4}$ ** Solution: **

Let P(n) be the given statement

P(n) = $1^{3}+2^{3}+3^{3}+ ........... + n^{3}$ = $\frac{n^{2}(n+1)^{2}}{4}$

**Step -1** : Let n = 1, L.H.S = $1^{3}$ = 1, R.H.S = $\frac{1^{2}(1+1)^{2}}{4}$ = 1

L.H.S = R.H.S ==> P(1) is true.

**Step -2** : Let P(n) be true for n = k

$1^{3}+2^{3}+3^{3}+ ........... + k^{3}$ = $\frac{k^{2}(k+1)^{2}}{4}$ --------- 1

We shall prove that P(k + 1) is true.

Add $(k+1)^{3}$ to both sides of equation (1)

$1^{3}+2^{3}+3^{3}+ ........... + k^{3}+ (k+1)^{3}$ = $\frac{k^{2}(k+1)^{2}}{4}$ + $(k+1)^{3}$

= $\frac{k^{2}(k+1)^{2}+4(k+1)^{3}}{4}$

= $\frac{(k+1)^{2}(k^{2}+4k+4)}{4}$

= $\frac{(k+1)^{2}(k+2)^{2}}{4}$

Thus we have

$1^{3}+2^{3}+3^{3}+ ........... + k^{3}+ (k+1)^{3}$ = $\frac{(k+1)^{2}(k+2)^{2}}{4}$

This statement is same as P(n) where n = k + 1

The statement P(n) is true for n = k + 1.

Thus P(k + 1) is true whenever P(k) is true.

**Step -3** : By Principle of Mathematical Induction P(n) is true for all natural integers.

**Question 2: **Prove by Mathematical Induction the sum of first n positive odd integers is $n^{2}$

** Solution: **

From the above given statement we have,

1 + 3 + 5 + 7+ .................. + (2n - 1) = $n^{2}$

Observe that (2n - 1) is the nth odd integer.

Let P(n) = 1 + 3 + 5 + 7+ .................. + (2n - 1) = $n^{2}$

**Step -1** : Let n = 1, L.H.S = 1, R.H.S = $1^{2}$ = 1

L.H.S = R.H.S ==>P(1) is true.

**Step -2** : Let P(n) be true for n = k

1 + 3 + 5 + 7+ .................. + (2k - 1) = $k^{2}$ -----------1

We shall prove that P(k + 1) is true.

Adding both sides (2k + 1) in equation (1)

1 + 3 + 5 + 7+ .................. + (2k - 1) + (2k - 1) = $k^{2} + (2k +1) $

1 + 3 + 5 + 7 + .................. + (2k - 1) + (2k - 1) = $(k + 1)^{2}$

This statement is same as P(n) where n = K + 1

The statement P(n) is true for n = k + 1.

Thus P(k + 1) is true whenever P(k) is true.

**Step -3** : By Principle of Mathematical Induction P(n) is true for all natural integers.