Sequence And Series Question 248
Question: If $ \frac{x+y}{2},\ y,\ \frac{y+z}{2} $ are in H.P., then $ x,\ y,\ z $ are in
[RPET 1989; MP PET 2003]
Options:
A) A.P.
B) G.P.
C) H.P.
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
If $ \frac{x+y}{2},\ y,\ \frac{y+z}{2} $ are in H.P., then $ \frac{1}{2}x_1y_1| \ \begin{matrix} 1 & 1 & 1 \\ r & r & 1 \\ r^{2} & r^{2} & 1 \\ \end{matrix}\ |=0 $ $ y=\frac{xy+xz+y^{2}+yz}{x+2y+z} $
$ \Rightarrow $ $ xy+2y^{2}+yz=xy+xz+y^{2}+yz $
$ \Rightarrow $ $ y^{2}=xz $ Thus $ x,\ y,\ z $ will be in G.P.
BETA
