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8. Sequences and Series
medium
If $b + c,$ $c + a,$ $a + b$ are in $H.P.$, then $\frac{a}{{b + c}},\frac{b}{{c + a}},\frac{c}{{a + b}}$ are in
A
$A.P.$
B
$G.P.$
C
$H.P.$
D
None of these
Solution
(a) Let $\frac{a}{{b + c}},\frac{b}{{c + a}},\frac{c}{{a + b}}$ are in $A.P.$
Add $1$ to each term, we get
$\frac{{a + b + c}}{{b + c}},\frac{{b + c + a}}{{c + a}},\frac{{c + a + b}}{{a + b}}$ are in $A.P.$
Divide each term by $(a + b + c),$
$\frac{1}{{b + c}},\frac{1}{{c + a}},\frac{1}{{a + b}}$ are in $A.P.$
Hence $b + c,\,\,c + a,\,\,a + b$ are in $H.P.$
which is given in question
Therefore, $\frac{a}{{b + c}},\frac{b}{{c + a}},\frac{c}{{a + b}}$ are in $A. P.$
Standard 11
Mathematics
