Sequence And Series Question 585
Question: The terms of a G.P. are positive. If each term is equal to the sum of two terms that follow it, then the common ratio is
Options:
A) $ \frac{\sqrt{5}-1}{2} $
B) $ \frac{1-\sqrt{5}}{2} $
C) 1
D) $ 2\sqrt{5} $
Show Answer
Answer:
Correct Answer: A
Solution:
Under condition $ T_{n}={T_{n+1}}+{T_{n+2}} $
$ \Rightarrow a{r^{n-1}}=ar^{n}+a{r^{n+1}}\Rightarrow {r^{n-1}}=r^{n}(1+r) $
$ \Rightarrow $ $ r^{2}+r-1=0\Rightarrow r=\frac{-1\pm \sqrt{1+4}}{2}=\frac{-1\pm \sqrt{5}}{2} $ Since, each term is $ +ve $ . Hence common ratio is $ \frac{\sqrt{5}-1}{2} $ .
BETA
