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itsnavin
September 12th, 2003, 09:41 PM
From a book, a number of pages are missing. The sum of the page numbers of these pages is 9808.

Which pages are missing?

amitrajora
September 13th, 2003, 03:00 PM
i saw the answer at
http://puzzle.dse.nl/harder/missing_pages_us.html

but how do you solve it ?

bnashier
September 13th, 2003, 06:41 PM
Dear Navin:

The book could have 10000 pages and only page no. 9808 could be missing and that will be a good answer as well. However, I know the intent of your question. It is a good question.
The question really is this: The sum of a certain number of consecutive positive integers is 9808. What are the numbers?

Solution: Assume that the numbers are k + 1, ............., k + n. Their sum is 1/2{n*2 + n(2k+1)}. Now the given condition gives you the equation
1/2{n*2 + n(2k+1)}= 9808 or n*2 + n(2k+1) = 19616 or n*2 + n(2k+1) - 19616 = 0. Now it is a quadratic equation in n and you learnt in your high school algebra class how to solve it. You need two numbers a & b such that ab = 19616 and a - b = 2k+1. The point to note here is that one of a & b is even and the other is odd. Factor 19616. You get 19616 = 32x613. Great ! 613 can’t be factored further ! This must be the odd number. Thus the numbers are a = 613 and b = 32. Then 2k + 1 = 613 - 32 = 581 gives you k = 290; and n = b = 32.

The numbers are 291, .........., 322.



Navin Farswal (Sep 12, 2003 12:11 p.m.):
From a book, a number of pages are missing. The sum of the page numbers of these pages is 9808.

Which pages are missing?

bnashier
September 14th, 2003, 02:15 AM
General solution:

Take two positive integers a & b of different parity, i.e., one is even and the other is odd. Then there is a sequence of successive integers whose sum is ab/2.

Say b is smaller of the two. Set m = (a-b-1)/2.
Then (m+1) + (m+2) + (m+3)+ ........+ (m+b) = mb + b(b+1)/2 = ab/2.

itsnavin
September 15th, 2003, 02:52 PM
Hello Budh ji
ur solution is perfect!...there was not much trick to solve the matematical equation but the key was problem formulation.

Amit: There were some other good puzzles on that site...good link

Cheers