Question

If $S_n$ denote the sum of $n$ terms of a G.P. whose first term is $a$$,andcommonratio$r$,findthesumof$S_{1}, S_{3}, S_{5}, ...S_{2n - 1}$.

Answer 1

Let $S_n$ be the sum of first $n$ terms of a G.P. and $r$ be common ratio
Formula for sum of $n$ terms of G.P. is given by
$S_n=\frac{a(r^n-1)}{r-1}$
Then, $S_1=\frac{a(r^1-1)}{r-1}$
$S_3=\frac{a(r^3-1)}{r-1}$
$S_5=\frac{a(r^5-1)}{r-1}$
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$S_{2n-1}=\frac{a(r^{2n-1}-1)}{r-1}$
Adding all terms, we get
$S=\frac{a}{r-1}\left[\right(r+r^3+.....+r^{2n-1})+(-1\left)\right(n\left)\right]$
Now, $r,r^3,r^5,....,r^{2n-1}$ represents G.P. with common ration $r^2$ and first term $r$
$\therefore S=\frac{a}{r-1}[\frac{r(r^{2n}-1)}{r^2-1}-n]$