# Q. One page is turn 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

(c) 9, 10

(d) 11, 12

Correct Answer: (b) 7, 8

#### Question from UPSC Prelims 2020 CSAT Paper

**Explanation : **

## Torn Pages Numbers

Let us assume that, the numbers written on the both sides of the missing page are x and (x + 1).

Since the sum of the numbers on the remaining pages is 195, we can set up the equation:

1 + 2 + 3 + … + (n-1) + n = 195 + x + (x+1)

We know that the sum of the first n natural numbers is:

1 + 2 + 3 + … + (n-1) + n = n(n+1)/2

Substituting this in our equation, we get:

- n(n+1)/2 = 195 + x + (x+1)
- n(n+1) = 2(195 + 2x + 1)
- n^2 + n = 2(2x + 196)
- n^2 + n = 4x + 392

Now we need to find which pair of consecutive numbers (x, x+1) will satisfy this equation. We can try each option:

For option (b) x = 7, n = 20, and this satisfies the equation.

Therefore, the torn page contains numbers 7 and 8, which is option (b).