Question

Find the sum of the series (3³ – 2³) + (5³ – 4³) + (7³ – 6³) + … to

(i) n terms

(ii) 10 terms

Answer 1

Let the series be S = (3³ – 2³) + (5³ – 4³) + (7³ – 6³) + ...

i) Generalizing the series in terms of i

⇒ S = 2n(n+1)(2n+1) + 3n(n+1) + n

⇒ S = 2n(2n² + 2n + n + 1) + 3n² + 3n + n

⇒ S = 4n³ + 6n² + 2n + 3n² + 4n

⇒ S = 4n³ + 9n² + 6n

Hence sum upto n terms is 4n³ + 9n² + 6n

ii) Sum of first 10 terms or upto 10 terms

To find sum upto 10 terms put n = 10 in S

⇒ S = 4(10)³ + 9(10)² + 6(10)

⇒ S = 4000 + 900 + 60

⇒ S = 4960

Hence sum of series upto 10 terms is 4960