Question

If for a given G.P. a=729 and 7th term is 64, determine $S_7$

Answer 1

$a=729t_7=64{ar}^6=64\Rightarrow r^6\times 729=64\Rightarrow r^6=\frac{64}{729}=6\sqrt{\frac{64}{729}}=\frac{2}{3}{S}_7=a+ar+a{r}^2+a{r}^3+a{r}^4+a{r}^5+a{r}^6=729+729\times \frac{2}{3}+........=\frac{a(1-r^n)}{(1-r)}=\frac{729(1-{\left(\frac{2}{3}\right)}^7)}{1-\frac{2}{3}}=\frac{729(1-\frac{64\times 2}{729\times 3})}{\frac{1}{3}}=\frac{(729\times 3-64\times 2)}{729\times 3}\times 729\times 3=2059$