Sum of Partial Sums of Geometric series

Question

Suppose we have a Geometric Sequence {a,r} a being the initial term and r being the common ratio. Is there a condensed formula for Sum of upto nth Partial sums of the terms in Geomtric series. $$ S = \sum_{i=0}^n ar^n + \sum_{i=0}^{n-1} ar^{(n-1)} +\sum_{i=0}^{n-2} ar^{(n-2)} .... + \ a $$

Answer 1

Sorry, What I meant was the following. Anyway I have done some formulation not sure if it is correct.

$$ S = \sum_{i=0}^n ar^i + \sum_{i=0}^{n-1} ar^i +\sum_{i=0}^{n-2} ar^i + .... + a $$ $$ = \frac{a(r^n -1)}{(r-1)} + \frac{a(r^{(n-1)} -1)}{(r-1)} + \frac{a(r^{(n-2)} -1)}{(r-1)} + .... + a $$

$$ = \frac{ar^n + ar^{(n-1)} + ar^{(n-2)} + .... + a(r-1) - na }{(r-1)} $$ $$ = \frac{a \frac{(r^n-1)}{(r-1)} - a(n+1)}{(r-1)} $$ $$ = \frac{a(r^n - 1) -a(n+1)(r-1))}{(r-1)^2} $$ $$ = \frac{a(r^n -(n+1)(r-1) -1)}{(r-1)^2} $$ $$ \frac{a(r^n-nr + n -r )}{(r-1)^2} $$

Answer 2

Sum of partial sums of geometric series: $$S=S_1+S_2+\cdots +S_{n-1}+S_n=\\ \frac{a(r-1)}{r-1}+\frac{a(r^2-1)}{r-1}+\cdots+\frac{a(r^{n-1}-1)}{r-1}+\frac{a(r^{n}-1)}{r-1}=\\ \frac{a(r+r^2+\cdots +r^{n-1}+r^n)-an}{r-1}=\\ \frac{a\cdot \frac{r(r^n-1)}{r-1}-an}{r-1}=\\ \frac{a(r^{n+1}-r-nr+n)}{(r-1)^2}$$ For example: $2,6,18,...$ $$S=S_1+S_2+S_3=2+(2+6)+(2+6+18)=36,\\ S=\frac{2(3^4-3-3^2+3)}{2^2}=\frac{144}{4}=36.$$