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 of terms be $n$.
Then ${57}^2-{13}^2=\left[n/2\right]\left[2(2m+1)(n-1).2\right]=n(2m+n)=2mn+n^2$
or ${57}^2-{13}^2={[n+m]}^2-m^2$
Hence $m=13andn+m=57orn=57-13=44$
Hence the required odd integers are $27,29,31,...,113.$