Question

If $l$ the last term, $d$ the common difference, and $s$ the sum of $n$ terms of an A.P. be connected by the equation $8ds=(d+2l{)}^2$, prove that $d=2a$.

Answer 1

Since the given relation is true for any number of terms, put $n=1$; then $a=l=s$.
Hence by substitution, $8ad=(d+2a{)}^2$,
or $(d-2a{)}^2=0$;
$\therefore d=2a$.