Question

Given a G.P. with $a=729$ and $7^{th}$ term $64$, determine $S_7$(Sum of first seven terms).

Answer 1

We have, $a=729,a_7=64$
Let $r$ be the common ratio of the G.P.
It is known that, $a_n=a{r}^{n-1}$
$\therefore a_7={ar}^{7-1}=\left(729\right)r^6\Rightarrow r^6=\frac{64}{729}\Rightarrow r^6={\left(\frac{2}{3}\right)}^6\Rightarrow r=\frac{2}{3}$
Also, it is known that $S_n=\frac{a(1-r^n)}{1-r}$
$\therefore S_7=\frac{729[1-{\left(\frac{2}{3}\right)}^7]}{1-\frac{2}{3}}=3\times 729[1-{\left(\frac{2}{3}\right)}^7]={\left(3\right)}^7\left[\frac{{\left(3\right)}^7-{\left(2\right)}^7}{{\left(3\right)}^7}\right]={\left(3\right)}^7-{\left(2\right)}^7=2187-128=2059$