Question

Given a G.P. with a = 729 and $7^{th}$ term 64. determine $S_7$

Answer 1

Here a = 729 and $a_7$ = 64

$\therefore a{r}^6=64\Rightarrow 729r^6=64$

$\Rightarrow r^6=\frac{64}{729}={\left(\frac{2}{3}\right)}^6\Rightarrow r=\frac{2}{3}$

Since $S_n=\frac{a(1-r^n)}{1-r}whenr<1$

$S_7=\frac{729[1-{\left(\frac{2}{3}\right)}^7]}{1-\frac{2}{3}}$

= $\frac{729[1-\frac{128}{2187}]}{\frac{3-2}{3}}$

= $729\times 3\left[\frac{2187-128}{2187}\right]$

= $\frac{729\times 3\times 2059}{2187}=2059.$