Question

The common ratio of a $G.P$ is $3$ and the last term is $486$. If the sum of these terms be $728$, find the first term

Answer 1

Given the common ratio of G.P$r=3$

$\Rightarrow$last term $l=a{r}^{n-1}=486$

$\Rightarrow a{3}^{n-1}=486$

$\Rightarrow \frac{3^{n}a}{3}=486$

$\Rightarrow 3^n.a=1458-\left(1\right)$

To find a

$\Rightarrow$Sum $=728$

$\Rightarrow$$\frac{a(r^n-1)}{r-1}=728$

$\Rightarrow \frac{a(3^n-1)}{3-1}=728$

$\Rightarrow a(3^n-1)=1456$

$\Rightarrow 3^n.a-a=1456$

From $\left(1\right)$ $\Rightarrow 1458-a=1456$

$\Rightarrow$$a=2$

$\therefore$ First term$=2$.