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.

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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.

Solution:

Given that,
The sum of GP $= 728$
Where, $r = 3, a = ?$
Firstly,
$T_{n}=a r^{n-1}$
$486=a 3^{n-1}$
$486=a 3^{n} / 3$
$486(3)=a 3^{n}$
$1458=a 3^{n} \ldots .$ Eq. (i)
$486=a 3^{n} / 3$ $486(3)=a 3^{n}$ $1458=a 3^{n} \ldots$ Eq.(i)
Sum of GP for $n$ terms $=a\left(r^{n}-1\right) /(r-1)$ $728=a\left(3^{n}-1\right) / 2$ $1456=a 3^{n}-a \ldots$ eq. $(2)$
Subtract eq. (1) from eq. (2)
$\begin{array}{l} 1458-1456=a \cdot 3^{n}-a \cdot 3^{n}+a \\ a=2 \end{array}$
Therefore, the first term is 2.