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.

It was those high school years, so long distant that I dealt with this type of thing therefore I am prepared to accept that I may well have misinterpreted.

Each line presents a separate calculation for the product of (x+j-1)  and (x+j).

Those products are progressive summed.  The bold line represents the only outcome that I can determine.  I note that if I increase x the Prog. sum moves further away from 10*n.


                                                             Prog.
x    n    j      x+j-1    x+j  Prod    10*n     Sum
1    1    1        1         2       2        10   
1    2    2        2         3       6        20          8
1    3    3        3         4     12        30         20
1    4   4       4         5     20       40         40
1    5    5        5         6     30        50         70

Posted by brianjn on 2011-02-22 20:27:02