If the sum of the first nth terms of a sequence is given by S_(n ) = 9(1 − (1/(3^n ))) (a) find the first and the second term of the sequence (b) find the nth term of the sequence (c) show that the sequence is a GP and find it common ratio please help.


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Question Number 6166 by sanusihammed last updated on 16/Jun/16

$I f t h e s u m o f t h e f i r s t n t h t e r m s o f a s e q u e n c e i s g i v e n b y$ $S_{n} = 9 (1 - \frac{1}{3^{n}})$ $(a) f i n d t h e f i r s t a n d t h e s e c o n d t e r m o f t h e s e q u e n c e$ $(b) f i n d t h e n t h t e r m o f t h e s e q u e n c e$ $(c) s h o w t h a t t h e s e q u e n c e i s a G P a n d f i n d i t c o m m o n r a t i o$ $p l e a s e h e l p .$
Commented by prakash jain last updated on 17/Jun/16

$First Term = S_{1} = 9 (1 - \frac{1}{3}) = 6$ $n^{t h} term = S_{n} - S_{n - 1} = 9 (1 - \frac{1}{3^{n}}) - 9 (1 - \frac{1}{3^{n - 1}})$ $= 9 (\frac{1}{3^{n - 1}} - \frac{1}{3^{n}}) = \frac{9 \cdot 2}{3^{n}} = \frac{18}{3^{n}}$ $ratio = \frac{a_{n}}{a_{n - 1}} = \frac{1}{3}$ $Since ratio is constant sequence is aGP$
Commented by sanusihammed last updated on 17/Jun/16

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