Q11 of 36 Page 161

Find the sum of the series

(33 – 23) + (53 – 43) + (73 – 63) + ... to (i) n terms (ii) 10 terms


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


i) Generalizing the series in terms of i



Using a3 – b3 = (a – b)(a2 + ab + b2)






We know that and



⇒ 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


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