Sequence And Series Question 354
Question: If $ a^{2},b^{2},c^{2} $ are in A.P. consider two statements
(i) $ \frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b} $ are in A.P. (ii) $ \frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b} $ are in A.P.
Options:
A) (i) and (ii) both correct
B) (i) and (ii) both correct
C) (i) correct (ii) incorrect
D) (i) incorrect (ii) correct
Show Answer
Answer:
Correct Answer: A
Solution:
Given $ a^{2},b^{2},c^{2} $ are in A.P.
$ \Rightarrow a^{2}+(ab+bc+ca),b^{2}+(ab+bc+ca) $ $ c^{2}+(ab+bc+ca) $ are in A.P.
$ \Rightarrow ,(a+b)(a+c),(b+c)(b+a),,(c+a)(c+b) $ are in AP.
$ \Rightarrow \frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b} $ are in A.P. [Divide by $ (a+b)(b+c)(c+a) $ ] Again, $ a^{2},b^{2},c^{2} $ are in A.P.
$ \Rightarrow \frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b} $ are in A.P.
$ \Rightarrow \frac{a+b+c}{b+c},\frac{a+b+c}{c+a},\frac{a+b+c}{a+b} $ are in A.P.
$ \Rightarrow \frac{a}{b+c}+1,\frac{b}{c+a}+1,\frac{c}{a+b}+1 $ are in A.P.
$ \Rightarrow \frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b} $ are in A.P.
BETA
