Consider the following statements : 1. The sum of 5 consecutive integers can be 100. 2. The product of three consecutive natural numbers can be equal to their sum. Which of the above statements is/are correct ?
- (a) 1 only
- (b) 2 only
- (c) Both 1 and 2 ✓ UPSC's answer
- (d) Neither 1 nor 2
Why the answer is (c)
• Let the five consecutive integers be n-2, n-1, n, n+1, n+2; their sum is 5n, so 5n = 100 implies n = 20, which is an integer, making Statement 1 correct.
• Let the three consecutive natural numbers be n, n+1, n+2; the equation for their product equaling their sum is n(n+1)(n+2) = 3n + 3.
• Testing small natural numbers: for n=1, the product is 1×2×3 = 6 and the sum is 1+2+3 = 6, which are equal, making Statement 2 correct.
• Since both Statement 1 and Statement 2 are mathematically valid, the correct option is (c).
Why the other options are wrong
- (a) 1 only
- Statement 2 is also correct because 1, 2, and 3 are consecutive natural numbers whose product (6) equals their sum (6).
- (b) 2 only
- Statement 1 is also correct because the sum of the consecutive integers 18, 19, 20, 21, and 22 is exactly 100.
- (d) Neither 1 nor 2
- Both statements are correct, as demonstrated by the integer solution n=20 for the sum and the natural number set {1, 2, 3} for the product-sum equality.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.