Determine all possible positive integer values of n that satisfy this equation:

Σ

j=n j=1

(x+j-1) (x+j) = 10*n

whenever x is an integer.

Note: For a desired value of n, the above equation holds true for at least one value of x.

According to my (clumsy to non-existent) powers of analysis:

1/3(n^3+3n^2x+3nx^2-n) = 10n
n^3/3+n^2x+nx^2-n/3 = 10n
n^2/3+nx+x^2-1/3=10
n^2+3nx+3x^2-1=30
n^2+3nx+3x^2 = 31

{n,x}={-11,5}{-11,6}{-7,1}{-7,6}{-4,-1}{-4,5}{4,-5}{4,1}{7,-6}{7,-1}{11,-6}{11,-5}

The positive integer values of n are 4,7,11, when x is a (positive or negative) integer.

Edited on February 23, 2011, 7:15 am

Posted by broll on 2011-02-23 07:08:40