CAPF AC 2022 Q1: General Mental Ability – Quantitative Aptitude
Which one of the following is the difference of the sum of cubes of first ten natural numbers and the sum of squares of first ten natural numbers?
- (a) 2400
- (b) 2640
- (c) 2880
- (d) 2000
Answer: (b) 2640
Explanation
Sum of cubes minus sum of squares
Given: find the difference between the sum of cubes of the first ten natural numbers and the sum of squares of the first ten natural numbers.
How to think: do not add ten cubes by hand.
Use the two standard formulas with n = 10.
| Quantity | Formula | Value for n = 10 |
|---|---|---|
| Sum of first n numbers | n(n + 1)/2 | 55 |
| Sum of cubes | [n(n + 1)/2]² | 55² = 3025 |
| Sum of squares | n(n + 1)(2n + 1)/6 | 10 × 11 × 21 ÷ 6 = 385 |
Difference: 3025 − 385 = 2640.
Answer: (b) 2640.
Why each option is right or wrong
- (a) Incorrect. 2400 is too small; the working gives 3025 − 385.
- (b) CORRECT. 3025 − 385 = 2640.
- (c) Incorrect. 2880 does not follow from either formula for n = 10.
- (d) Incorrect. 2000 is a rough guess; the exact difference is 2640.
Key point
Σn = n(n+1)/2. Σn² = n(n+1)(2n+1)/6. Σn³ = [n(n+1)/2]². For n = 10: 55, 385 and 3025.
Remember
First ten numbers: sum 55, squares 385, cubes 3025.
Common mistakes
- Forgetting that the sum of cubes is simply the square of the sum of the numbers.
- Using 2n − 1 instead of 2n + 1 in the formula for the sum of squares.
- Subtracting the sum of the numbers (55) instead of the sum of squares (385).
Exam tip
Educated guess: 3025 ends in 5 and 385 ends in 5, so the difference must end in 0. All options do, so use a second check: 3025 − 385 is a little under 2700, which points to 2640.
Sources
- NCERT Class XI Mathematics — Sequences and Series (sum to n terms of special series)
- R. S. Aggarwal, Quantitative Aptitude — Number System and Series