CAPF AC 2021 Q79: General Mental Ability – Quantitative Aptitude
Suppose the nth term of a series is 1 + n/2 + n²/2. If there are 20 terms in the series, then the sum of the series is equal to
- (a) 1360
- (b) 1450
- (c) 1500
- (d) 1560
Answer: (d) 1560
Explanation
Sum of a series from its nth term
Given: the nth term is 1 + n/2 + n²/2 and there are 20 terms; find the sum.
How to think: split the term into three parts and add each part separately for n = 1 to 20.
Use the standard sums of n and n².
| Part | Formula for n = 1 to 20 | Value |
|---|---|---|
| Sum of 1 | 20 × 1 | 20 |
| Sum of n | 20 × 21 ÷ 2 | 210 |
| Sum of n² | 20 × 21 × 41 ÷ 6 | 2870 |
Put together: Sum = 20 + 210/2 + 2870/2.
That is 20 + 105 + 1435 = 1560.
Answer: (d) 1560.
Why each option is right or wrong
- (a) Incorrect. 1360 is 200 short; the working gives 20 + 105 + 1435.
- (b) Incorrect. 1450 comes from an error in the sum of squares.
- (c) Incorrect. 1500 is close but leaves out part of the sum.
- (d) CORRECT. 20 + 105 + 1435 = 1560.
Key point
Sum of first n natural numbers = n(n+1)/2. Sum of their squares = n(n+1)(2n+1)/6. For n = 20 these are 210 and 2870.
Remember
Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6.
Common mistakes
- Forgetting to halve the sums of n and n², since the term has n/2 and n²/2.
- Adding only the 20th term instead of all twenty terms.
- Using the formula n(n+1)/2 for the squares as well.
Exam tip
Educated guess: the 20th term alone is 1 + 10 + 200 = 211, so the total is well over 1000. With close options the full working is needed; split the term into three easy sums.
Sources
- NCERT Class XI Mathematics — Sequences and Series (sum to n terms of special series)
- NCERT Class X Mathematics — Arithmetic Progressions