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

Here, common ratio, r = 3
n^th term, a_n = 486
S_n = 728

$$\begin{gathered} a_n=486 \\ \Rightarrow a{r}^{n-1}=486 \\ \Rightarrow a{\left(3\right)}^{n-1}=486 \\ \Rightarrow a{\left(3\right)}^n=486\times 3 \\ \Rightarrow a{\left(3\right)}^n=1458...\left(\mathrm{i}\right) \\ \mathrm{Now},S_n=728 \\ \Rightarrow 728=a\left(\frac{3^n-1}{3-1}\right) \\ \Rightarrow 728=\left\{\frac{a{\left(3\right)}^n-a}{2}\right\} \\ \Rightarrow 1456=a{\left(3\right)}^{n-1}-a \\ \Rightarrow 1456=1458-a\left[\mathrm{From}\left(\mathrm{i}\right)\right] \\ \Rightarrow a=1458-1456 \\ \Rightarrow a=2 \end{gathered}$$