## Question-68

**Question-68**

Let P(n) be the statement that 1^{3 } + 2^{3} + …. + n^{3} = for the positive integer n.

**a)** What is the statement P(1) ?

P(1) is the statement

**b) ** Show that P (1) is true, completing the basis step of the proof.

The left-hand side of the basis step is ?

The right- hand side of the basis step is ?

**c) ** What is the inductive hypothesis?

The inductive hypothesis is the statement that 1^{3} + 2^{3} + ….+ k^{3} = ?

**d)** What do you need to prove in the inductive step?

You need to prove that 1 + 2 +….. + k + = ?

**e)** Complete the inductive step, identifying where you use the inductive hypothesis.

**Solution**

**a)** .

**b)**

The left-hand side of the basis step is = 1.

The right- hand side of the basis step is = 1.

**c)**.

**d) **.

For all k 1.

**e) **Ans The inductive hypotheis replaces the quantity 1^{3} + 2^{3} + …. + k^{3} from the left-hand side of part (d) , which shows that [1^{3} + 2^{3} + … + k^{3} ] + (k + 1)^{3.}

^{ = }