Sequence And Series Question 287
Question: If a ,b, c are in A.P., then $ \frac{1}{\sqrt{a}+\sqrt{b}},,\frac{1}{\sqrt{a}+\sqrt{c}}, $ $ \frac{1}{\sqrt{b}+\sqrt{c}} $ are in
[Roorkee 1999; Kerala (Engg.) 2005]
Options:
A) A.P.
B) G.P.
C) H.P.
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
a, b, c are in A.P. i.e., 2b = a + c Let $ \frac{1}{\sqrt{a}+\sqrt{c}}-\frac{1}{\sqrt{a}+\sqrt{b}}=\frac{1}{\sqrt{b}+\sqrt{c}}-\frac{1}{\sqrt{a}+\sqrt{c}} $
Þ $ \frac{\sqrt{b}-\sqrt{c}}{\sqrt{a}+\sqrt{b}}=\frac{\sqrt{a}-\sqrt{b}}{\sqrt{b}+\sqrt{c}} $
$ \Rightarrow b-c=a-b $
Þ $ 2b=a+c $
$ \therefore \frac{1}{\sqrt{a}+\sqrt{b}},\frac{1}{\sqrt{a}+\sqrt{c}},\frac{1}{\sqrt{b}+\sqrt{c}} $ are in A.P.
BETA
