Question 725436
If you have studied polynomials and/or quadratic equations,
{{{n^2+n=240}}} <--> {{{n^2+n-240=0}}}
At that point you would solve either
by factoring,
or by "completing the square",
or by using the quadratic formula.
 
However, it is easier to solve {{{n(n+1)=240}}}
by finding two consecutive numbers whose product is 240.
 
You could find factors from 1 up, and they would come in pairs:
{{{240=1*240}}}
{{{240=2*120}}}
{{{240=3*80}}}
{{{240=4*60}}}
{{{240=5*48}}}
{{{240=6*40}}}
7 is not a factor
{{{240=8*30}}}
9 is not a factor
{{{240=10*24}}}
and so on.
 
But if you have some factoring practice, the answer will jump at you.
It is obvious that {{{240=8*30}}, right?
{{{240=8*30}}}={{{(8*2)*(30/2)=16*15}}} is the product of two consecutive numbers:
{{{highlight(n=15)}}} and {{{highlight(n+1=16)}}}
{{{240=16*15}}} is what you would need to figure out to solve {{{n^2+n-240=0}}} by factoring, anyway.