Sequence And Series Question 387
Question: If $ x=\sum\limits_{n=0}^{\infty }{a^{n}},y=\sum\limits_{n=0}^{\infty }{b^{n}},,z=\sum\limits_{n=0}^{\infty }{c^{n}} $ where a, b, c are in A.P and $ | a |<1,| b |<1,| c |<1 $ then $ x,y,z $ are in
Options:
A) G.P.
B) A.P.
C) H.P.
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ x=\sum\limits_{n=0}^{\infty }{a^{n}}=\frac{1}{1-a};a=1-\frac{1}{x} $ $ y=\sum\limits_{n=0}^{\infty }{b^{n}}=\frac{1}{1-b};,b=1-\frac{1}{y} $ $ z=\sum\limits_{n=0}^{\infty }{c^{n}}=\frac{1}{1-c};c=1-\frac{1}{z} $ a, b, c care in A.P. OR $ 2b=a+c $ $ 2( 1-\frac{1}{y} )=1-\frac{1}{x}+1-\frac{1}{y}\frac{2}{y}=\frac{1}{x}+\frac{1}{z} $
$ \Rightarrow x,y,z $ are in H.P.
BETA
