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.
Solution:
Here, common ratio, r = 3
nth term, an = 486
Sn = 728
$a_{n}=486$
$\Rightarrow a r^{n-1}=486$
$\Rightarrow a(3)^{n-1}=486$
$\Rightarrow a(3)^{n}=486 \times 3$
$\Rightarrow a(3)^{n}=1458$ ....(i)
Now, $S_{n}=728$
$\Rightarrow 728=a\left(\frac{3^{n}-1}{3-1}\right)$
$\Rightarrow 728=\left\{\frac{a(3)^{n}-a}{2}\right\}$
$\Rightarrow 1456=a(3)^{n-1}-a$
$\Rightarrow 1456=1458-a$ [From (i)]
$\Rightarrow a=1458-1456$
$\Rightarrow a=2$
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