Question 894823
n and n+1 are the consecutive integers.

{{{highlight_green(1/n+1/(n+1)=25/156)}}}


Lowest common denominator is 156n(n+1); so multiply both sides by this.

{{{156(n+1)+156n=25n(n+1)}}}
{{{156n+156+156n=25n^2+25n}}}
{{{312n+156=25n^2+25n}}}
{{{25n^2+25n=312n+156}}}
{{{2n^2=287n+156}}}---THE MISTAKE HERE!!!
{{{highlight_green(25n^2-287n-156=0)}}}---should be factorable, but not worth the effort.


{{{n=(287+- sqrt(287^2+4*25*156))/(2*25)}}}
{{{n=(287+- sqrt(97969))/(2*25)}}}
<s>NO.  Not sensible.  83617 is not a square number.  n will not be an integer.</s>


Solution that works:  <b>The numbers are 12 and 13.</b>