SOLUTION: The pages of a report are numbered consecutively from 1 to 10. If the sum of the page numbers up to and including page number x of the report is equal to one more than the sum of t

Algebra ->  Functions -> SOLUTION: The pages of a report are numbered consecutively from 1 to 10. If the sum of the page numbers up to and including page number x of the report is equal to one more than the sum of t      Log On


   

Question 3461: The pages of a report are numbered consecutively from 1 to 10. If the sum of the page numbers up to and including page number x of the report is equal to one more than the sum of the page numbers following page number x, then x =
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8
Answer by drglass(89) About Me  (Show Source):
You can put this solution on YOUR website!
To do this problem it is helpful to know that the sum of the numbers from 1 to x is %28n%28n+%2B+1%29%29%2F2. This means the sum of the page numbers from 1 to x is:

%28x%28x+%2B+1%29%29%2F2.

To find the remaining sum, think about it this way 5+%2B+4+=+1+%2B+2+%2B+3+%2B+4+%2B+5+-+%281+%2B+2+%2B+3%29. In other words the sum from (x+1) to 10 is the sum of 1 to 10 minus the sum of 1 to x. This allows us to write a relatively simple expression for the sum from (x+1) to 10

%2810%2810+%2B+1%29%29%2F2+-+%28x%28x+%2B+1%29%29%2F2

Since the sum from one to x is 1 plus the sum of x+1 to 10, we get the following equation

or

%28x%28x+%2B+1%29%29%2F2+=+56++-+%28x%28x+%2B+1%29%29%2F2

If we add %28x%28x+%2B+1%29%29%2F2 to both sides we get

x%28x+%2B+1%29+=+56

The equation becomes x%5E2+%2B+x+-+56+=+0 which factors nicely to %28x+-+7%29%28x+%2B+8%29+=+0. So the solution to the equation is x = 7 and x = -8. Well, you cannot have negative pages, so the answer to the question must be 7. Let's check it.

the sum from 1 to 7 is %287%287%2B1%29%29%2F2+=+28 and 8+%2B+9+%2B+10+=+27, which is one more than the sum from 1 to 7. So the answer is indeed 7.