one page is torn form 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?
Answers
Step-by-step explanation:
Let x be the (odd) number of the page that is torn out.
Then 2x+1 is the sum of the page numbers on the page that is torn out.
Let n be the total number of pages.
The formula for the sum of numbers in a series from 1 ..n is [math]\frac{n(n+1)}{2}[/math]
So
[math]\frac{n(n+1)}{2} - (2x+1) = 525[/math]
→
[math]\frac{n(n+1)}{2} - 2x - 1 = 525[/math]
→
[math]\frac{n(n+1)}{2} -2x -526 = 0[/math]
Multiplying by 2 and doing some rearranging yields
[math]0 = n^{2}+n - 4x -1052[/math]
Now, the key to solving this problem is to remember that
[math](n+a)(n-b) = n^{2}+(a-b)n-a*b[/math]
In this case, the coefficient of the linear term (n) is 1. In order to make that happen, a and b need to be two integers whose difference is 1 and such that
[math]-a*b = -4x -1052 [/math] → [math]4x = a*b - 1052[/math]
where x is a positive integer. The obvious place to start looking for consecutive integers that multiply to make a certain number is the square root:
[math]\sqrt{1052} = 32.43[/math] which tells us to try out a = 33 and b = 32.
Sure enough, 32*33 = 1056, so
[math]4x = 1056–1052 = 4 [/math] → [math]x=1[/math]
Now, remember that we are looking for the sum of the numbers on the removed page, which is [math]2x+1=3.[/math]