Question

Sum of certain consecutive odd positive integers is ${57}^2-{13}^2$. Find them.

Answer 1

Let the odd integers be $2m+1,2m+3,2m+5$,_______ and let their number be n. Then
${57}^2-{13}^2=\left(n/2\right)\left[2\right(2m+1)+(n-1\left).2\right]$
${57}^2-{13}^2=(n+m{)}^2-m^2$
Hence $m=13$ and $n+m=57$
or $n=57-13-44$
Hence the required odd integers are $27,29,31,$____,$113$.