Sequence And Series Question 24
Question: If the sum of $ n $ terms of a G.P. is 255 and $ n^{th} $ terms is 128 and common ratio is 2, then first term will be
[RPET 1990]
Options:
A) 1
B) 3
C) 7
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Given that $ \frac{a(r^{n}-1)}{r-1}=255 $ $ (\because \ \ r>1) $ ?..(i) $ a{r^{n-1}}=128 $ ?..(ii) and common ratio $ r=2 $ ?..(iii) From (iii), (i) and (ii) we get $ a_1=h_1=2,\ a_{10}=h_{10}=3 $ ?..(iv) and $ \frac{a(2^{n}-1)}{2-1}=255 $ …..(v) Dividing (v) by (iv) we get $ \frac{2^{n}-1}{{2^{n-1}}}=\frac{255}{128} $
$ \Rightarrow $ $ 2-{2^{-n+1}}=\frac{255}{128} $
$ \Rightarrow $ $ {2^{-n}}={2^{-8}} $
$ \Rightarrow $ $ n=8 $ Putting $ n=8 $ in equation (iv), we have $ a\ .\ 2^{7}=128=2^{7} $ or $ a=1 $ .
BETA
