Question

What is the sum of the first 50 terms of the series (1 × 3) + (3 × 5) + (5 × 7) + ...?

विकल्प

• 1,71,650

• 26,600

• 26,650

• 26.900

Answer 1

1,71,650

Explanation:

The given series is (1 × 3) + (3 × 5) + (5 × 7) + ..........

Its general term is given by T_n = (2n  – 1)(2n + 1) = 4n² – 1

Sum of series = `4sumn^2 - sum1`

S_n = `(4n(n + 1)(2n + 1))/6 - n`

S_n = `n[(2(2n^2 + 3n + 1))/3 - 1]`

S_n = `n[(4n^2 + 6n + 2 - 3)/3] = [(n(4n^2 + 6n - 1))/3]`

For sum of first 50 terms of the series,

n = 50, ⇒ S₅₀ = `(50[4(50)^2 + 6(50) - 1])/3`

= `(50 xx (10000 + 300 - 1))/3`

= `(50 xx 10299)/3`

= 171650