The first term of a $G P$ is $243$ and the fifth term is $48$ , find the common ratio and the sum of first $6$ terms.

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Solution

$a = 243$
$a_{5} = a r^{(} n - 1) = a r^{4} = 48$
$= 243 r^{4} = 48$
$r^{4} = (\frac{2}{3})^{4}$
$r = \frac{2}{3}$
now sum of six trems $= S_{6} = \frac{243 (1 - r^{6})}{1 - r}$
$S_{6} = \frac{243 (1 - (\frac{2}{3})^{6})}{1 - \frac{2}{3}} S_{6} = 243 (1 - (\frac{64}{729})^{6}) 3 s_{6} = 665$

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