Sequence And Series Question 2
Question: If every term of a G.P. with positive terms is the sum of its two previous terms, then the common ratio of the series is
[RPET 1986]
Options:
A) 1
B) $ \frac{2}{\sqrt{5}} $
C) $ \frac{\sqrt{5}-1}{2} $
D) $ \frac{\sqrt{5}+1}{2} $
Show Answer
Answer:
Correct Answer: D
Solution:
Let first term and common ratio of G.P. are respectively $ a $ and $ r $ , then under condition, $ T_{n}={T_{n-1}}+{T_{n-2}} $
$ \Rightarrow $ $ a{r^{n-1}}=a{r^{n-2}}+a{r^{n-3}} $
$ \Rightarrow $ $ a{r^{n-1}}=a{r^{n-1}}{r^{-1}}+a{r^{n-1}}{r^{-2}} $
$ \Rightarrow $ $ 1=\frac{1}{r}+\frac{1}{r^{2}} $
$ \Rightarrow $ $ r^{2}-r-1=0 $
$ \Rightarrow $ $ r=\frac{1\pm \sqrt{1+4}}{2}=\frac{1+\sqrt{5}}{2} $ Taking only (+) sign $ (\because \ r>1) $ .
BETA
