Sequence And Series Question 410
Question: The sum of an infinite GP is x and the common ratio r is such that $ | r |<1 $ . If the first term of the GP is 2, then which one of the following is correct?
Options:
A) $ -1<x<1 $
B) $ -\infty <x<1 $
C) $ 1<x<\infty $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
[c] GP = x $ \frac{a}{1-r}=x $ (where, a = 1st term and r = common ratio)
$ \Rightarrow \frac{2}{1-r}=x $ … (i) ( $ \because $ Given $ a=2 $ and $ |r|<1 $ )
$ \Rightarrow ,-1<r<1\Rightarrow 1>-r>-1 $
$ \Rightarrow 1+1>1-r>1-1 $
$ \Rightarrow ,0<1-r<2 $
$ \Rightarrow ,\frac{1}{1-r}>\frac{1}{2},\frac{2}{1-r}>1 $ from equation (i) $ x>1 $ Hence, $ 1<x<\infty . $
BETA
