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Question
Find the sum of the series (33 – 23) + (53 – 43) + (73 – 63) + ... to 10 terms
Solution
Given series
⇒ (33 – 23) + (53 – 43) + (73 – 63) + ...
= (33 + 53 + 73 + …) – (23 + 43 + 63 + …)
⇒ [33 + 53 + 73 + … (2n + 1)3] – [23 + 43 + 63 + … (2n)3]
∴ Tn = (2n + 1)3 – (2n)3
= (2n + 1 – 2n) [2n + 1)2 + (2n + 1)(2n) + (2n)2] ....[∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= 1 · [4n2 + 1 + 4n + 4n2 + 2n + 4n2]
= 12n2 + 6n + 1
S10 = 4(10)3 + 9(10)2 + 6(10)
= 4 × 1000 + 900 + 60
= 4000 + 960
= 4960
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