Suppose the n^th term of a series is \(1+\frac{n}{2}+\frac{n^{2}}{2} \). If there are 20 terms in the series, then the sum of the series is equal to

Question

Question

This question was previously asked in

UPSC CAPF Paper - I: Official Paper (Held on: 8 Aug 2021)

Answer 1

Given:

nth term = 1 + n/2 + n²/2

Sum of natural numbers = n(n+1)/2

Sum of square of 'n' natural numbers = [n(n+1)(2n+1)]/6

Calculation:

For understanding, breaking the terms given in question as :

1. Adding one twenty times = 20

2. When n = 1, 2, 3 and so on up to 20 in n/2, the sum = 1/2 (n(n+1)/2)

= 1/2 (20(20+1)/2) = 105

3. When n = 1, 2, 3 and so on up to 20 in n²/2, the sum = 1/2 [n(n+1)(2n+1)]/6

= 1/2 [20(20+1)(2×20+1)]/6 = 1435

Therefore, sum of the series = 20 + 105 + 1435 = 1560

Hence, option 4 is correct.

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