Question

By the principle of mathematical induction, prove the following:

1³ + 2³ + 3³ + ….. + n³ = `("n"^2("n + 1")^2)/4` for all x ∈ N.

Answer 1

Let P(n) be the statement 1³ + 2³ + 3³ + ….. + n³ = `("n"^2("n + 1")^2)/4` for all x ∈ N.

i.e., p(n) = 1³ + 2³ + …… + n³ = `("n"^2("n + 1")^2)/4` for all x ∈ N

Put n = 1

LHS = 1³ = 1

RHS = `(1^2(1 + 1)^2)/4`

`= (1 xx 2^2)/4`

`= 4/4` = 1

∴ P(1) is true.

Assume that P(n) is true n = k

P(k): 1³ + 2³ + …… + k³ = `(k^2(k + 1)^2)/4`

To prove P(k + 1) is true.

i.e., to prove 1³ + 2³ + ……. + k³ + (k + 1)³ = `((k + 1)^2 ((k+1)+1)^2)/4 = ((k + 1)^2(k + 2)^2)/4`

Consider 1³ + 2³ + …… + k³ + (k + 1)³ = `(k^2(k + 1)^2)/4 + (k + 1)^3`

= (k + 1)² `[k^2/4 + (k + 1)]`

= (k + 1)² `[(k^2 + 4(k + 1))/4]`

`= ((k + 1)^2(k + 2)^2)/4`

⇒ P(k + 1) is true, whenever P(k) is true.

Hence, by the principle of mathematical induction P(n) is true for all n ∈ N.