Question

For the following GP.s, find $\rm S_{n}$,
i) 2, 6, 18, 54, ...
ii) $\rm a,b,\frac{b^{2}}{a}, \frac{b^{3}}{a^{2}},...$

Answer 1

Solution:

i) 2, 6, 18, 54, ...

here a= 2

common ratio r= 3

To find the Sum of n terms of this GP,since here r > 1,
$S_{n} = \frac{a( {r}^{n} - 1)}{r - 1} \\ \\ S_{n}= \frac{2( {3}^{n} - 1)}{3 - 1} \\ \\ S_{n}= \frac{2( {3}^{n} - 1)}{2} \\ \\ S_{n}= ( {3}^{n} - 1)$
ii) $\rm a,b,\frac{b^{2}}{a}, \frac{b^{3}}{a^{2}},...$

here first term a

common ratio r = b/a

$S_{n} = \frac{a( {r}^{n} - 1)}{r - 1} \\ \\ S_{n}= \frac{a[ { (\frac{b}{a}) }^{n} - 1]}{ \frac{b}{a} - 1} \\ \\ S_{n}= \frac{ {a}^{2} ( {b}^{n} - {a}^{n} )}{ {a}^{n} (b - a)} \\ \\ S_{n}= \frac{ {a}^{2 - n} ( {b}^{n} - {a}^{n} )}{(b - a)}$

Hope it helps you