Prove by the method of induction, for all n ∈ N. 1^2 + 3^2 + 5^2+....+ (2n - 1)^2 = n/3 (2n-1)(2n+1)

Question

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

1² + 3² + 5²+....+ (2n - 1)² = \(\frac{n}3(2n-1)(2n+1)\)

Answer 1

Let P(n) = 1² + 3² + 5² +…..+ (2n – 1)² = \(\frac{n}3\)(2n – 1)(2n + 1), for all n ∈ N.

Step I: Put n = 1

 L.H.S. = 1² = 1

R.H.S. = \(\frac13\)[2(1) – 1][2(1) + 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.

∴ 1² + 3² + 5² +….+(2k – 1)² = \(\frac{k}3\)(2k – 1)(2k + 1) …….(i)

Step III:

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

i.e., to prove that

= R.H.S

∴ 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.

∴ 1² + 3² + 5² + …+ (2n – 1)² = \(\frac{n}3\)(2n – 1)(2n + 1) for all n ∈ N