Question

Prove by the method of induction, for all n ∈ N.

1³ + 3³ + 5² + ...+(2n - 1)² = n²(2n² - 1)

Answer 1

Let P(n) = 1³ + 3³ + 5³ + …. to n terms = n² (2n² – 1), for all n ∈ N.

But 1, 3, 5, are in A.P.

∴ a = 1, d = 2

Let t_n be the nth term.

t_n= a + (n – 1) d = 1 + (n – 1) 2 = 2n – 1

∴ P(n) = 1³ + 3³ + 5³ +…..+ (2n – 1)³ = n (2n² - 1)

Step I:

Put n = 1

L.H.S. = 1³ = 1

R.H.S. = 1² [2(1)² – 1] = 1

∴ L.H.S. = R.H.S.

∴ P(n) is true for n = 1

Step II: Let us assume that P(n) is true for n = k.

∴ 1³ + 3³ + 5³ +…+ (2k – 1)³ = k² (2k² – 1) …(i)

Step III: We have to prove that P(n) is true for n = k + 1,

i.e., to prove that

∴ P(n) is true for n = k + 1.

Step IV: From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

∴ 1³ + 3³ + 5³ + … to n terms = n² (2n² – 1) for all n ∈ N.