A student’s exercise booklet has its pages numbered from 1 onwards in the standard format. Due to damage, one double-sided sheet got detached, which had two consecutive page numbers. The total of all visible page numbers now adds up to 290. Which of the following pairs of page numbers were likely on the missing sheet?
(a) 11, 12
(b) 15, 16
(c) 19, 20
(d) 17, 18

Question

A student&rsquo;s exercise booklet has its pages numbered from 1 onwards in the standard format. Due to damage, one double-sided sheet got detached, which had two consecutive page numbers. The total of all visible page numbers now adds up to 290. Which of the following pairs of page numbers were likely on the missing sheet?
(a) 11, 12
(b) 15, 16
(c) 19, 20
(d) 17, 18

ONLiNE UPSC · Accepted Answer

Let&rsquo;s assume the total number of pages in the booklet is n, and no other pages are missing.
Sum of first n natural numbers =
S = n * (n+1)/2
We're told that after removing one sheet (2 pages), the remaining visible pages sum to 290. So:
S&minus;(missing pages) = 290
Try small values of n and see when the total sum minus a pair of consecutive (odd, even) pages gives 290.
Try n=25:
S = (25 * 26)/2
Missing sum = 325 - 290 = 35
And since 17 is odd and 18 is even &mdash; this is a valid double-sided sheet.
As, 17 + 18 = 35
Hence, option d is correct.

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M Question 1	⏱ 0
A student’s exercise booklet has its pages numbered from 1 onwards in the standard format. Due to damage, one double-sided sheet got detached, which had two consecutive page numbers. The total of all visible page numbers now adds up to 290. Which of the following pairs of page numbers were likely on the missing sheet? (a) 11, 12 (b) 15, 16 (c) 19, 20 (d) 17, 18