Question

First term of a GP is 729 and 7th term is 64. Find the sum of first seven terms.

Answer 1

$S_{n}ofGP=\frac{a(1-r^n)}{1-r}$
$t_n-n^{th}$ term of GP $=a{r}^{n-1}$
Here a = 729 $t_n=t_7=a{r}^6=64$
$\therefore a{r}^6=64$ or $r^6=\frac{64}{a}=\frac{64}{729}=\frac{2^6}{3^6}={\left(\frac{2}{3}\right)}^6$
$\therefore r=\frac{2}{3}$
$S_7=\frac{729(1-\frac{2}{3}{)}^7}{1-\frac{2}{3}}=\frac{729\{1-\frac{128}{2187}\}}{\frac{1}{3}}=729\times 3-\frac{729\times 3\times 128}{2187}=2187-128=2059$