Question

The difference between fourth and first term of a $G.P.$ is $52$ and sum of the first three terms is $26$. If $T_n$ and $S_n$ represent $n^{th}$ term and sum to $n$ terms respectively of the $G.P.$, then which of the following is/are true

• A. $T_7=486$
• B. $T_7=1458$
• C. $S_6=972$
• D. $S_6=728$

Answer 1

Answer: (B), (D)

$S_6=728$

Let the $G.P.$ be $a,ar,a{r}^2,....$
Given $a{r}^3-a=52...\left(1\right)$
$a+ar+a{r}^2=26...\left(2\right)$

Dividing equation $\left(1\right)$ by $\left(2\right)$, we get
$\frac{r^3-1}{1+r+r^2}=2$
$\Rightarrow r-1=2\Rightarrow r=3$

Substituting value of $r$ in equation $\left(1\right)$
$26a=52\Rightarrow a=2$

$\therefore T_7=2\times 3^6=1458\Rightarrow S_6=\frac{2(3^6-1)}{3-1}=728$