One page is torn from a booklet whose pages are numbered in the usual manner starting from the first page as 1. The sum of the numbers on the remaining pages is 195. The torn page contains which of the following numbers?
- (a) 5, 6
- (b) 7, 8 ✓ UPSC's answer
- (c) 9, 10
- (d) 11, 12
Why the answer is (b)
• A torn leaf carries two consecutive page numbers, odd then even: (2k − 1, 2k), whose sum is 4k − 1.
• If the booklet has n pages, the total is n(n + 1)/2, so n(n + 1)/2 − (4k − 1) = 195.
• n = 20 gives a total of 210, so the torn pages add up to 210 − 195 = 15, i.e. pages 7 and 8.
• n = 19 gives 190 (less than 195) and n = 21 gives 231, needing pages that sum to 36, which is not of the form 4k − 1; so n = 20.
• Hence option (b).
Why the other options are wrong
- (a) 5, 6
- The sum of 5 and 6 is 11, which would imply a total sum of 206, not a triangular number.
- (c) 9, 10
- The sum of 9 and 10 is 19, which would imply a total sum of 214, not a triangular number.
- (d) 11, 12
- The sum of 11 and 12 is 23, which would imply a total sum of 218, not a triangular number.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.