Question

If $S_1,S_2,S_3,.....S_r$ are the sums of $n$ terms of arithmetic series whose first terms are $1,2,3,4,....;$ and whose common differences are $1,3,5,7,....;$ find the value of $S_1+S_2+S_3+....+S_r$.

Answer 1

We have $S_1=\frac{n}{2}\{2+(n-1\left)\right\}=\frac{n(n+1)}{2}$,
$S_2=\frac{n}{2}\{4+(n-1\left)3\right\}=\frac{n(3n+1)}{2}$,
$S_3=\frac{n}{2}\{6+(n-1\left)5\right\}=\frac{n(5n+1)}{2}$,
$S_r=\frac{n}{2}\{2p+(n-1\left)\right(2p-1\left)\right\}=\frac{n}{2}\left\{\right(2p-1)n+1\}$;
$\therefore$ The require $sum=\frac{n}{2}\left\{\right(n+1)+(3n+1)+.....(\overline{2p-1}.n+1\left)\right\}$
$=\frac{n}{2}\left\{\right(n+3n+5n+....\overline{2p-1}.n)+p\}$
$=\frac{n}{2}\left\{n\right(1+3+5+....\overline{2p-1})+p\}$
$=\frac{n}{2}(n{p}^2+p)$
$=\frac{np}{2}(np+1)$.