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-th

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

T h a n k s s o m u c h