Question

If first term of a G.P. is 729 and its seventh term is 64 , then sum of first seven terms is

• A. 2080
• B. 2060
• C. 2059
• D. 2120

Answer 1

Answer: (C)

Given, $a=729,T_7=64$
$\Rightarrow a{r}^{7-1}=64\left(\because T_n=a{r}^{n-1}\right)$
$\Rightarrow 729r^6=64$
$\Rightarrow r^6=\frac{64}{729}={\left(\frac{2}{3}\right)}^6$
On comparing base of the power 6 on both sides, we get
$\Rightarrow r=\frac{2}{3}<1$
Now, $S_n=\frac{a\left(1-r^n\right)}{1-r}$
$(\because r<1)$
$\therefore S_7=\frac{729\left[1-{\left(\frac{2}{3}\right)}^7\right]}{1-\frac{2}{3}}=\frac{729\left[1-\frac{2^7}{3^7}\right]}{\frac{1}{1}-\frac{2}{3}}$
$=\frac{729\left[\frac{1}{1}-\frac{128}{2187}\right]}{\frac{3-2}{3}}$
$=\frac{729\times 3}{1}\times \frac{2187-128}{2187}$
$=\frac{1}{1}\times 2059$
$=2059$