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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 94
Total: 106
If a(
1
b
+
1
c
) , b(
1
c
+
1
a
) , c(
1
a
+
1
b
) are in A.P., prove that a, b, c are in A.P.
Solution:  
We have, a(
1
b
+
1
c
) , b(
1
c
+
1
a
) , c(
1
a
+
1
b
) are in A.P.
⇒
a(b+c)
bc
 ,
b(a+c)
ca
 ,
c(a+b)
ab
 are in A.P.
⇒
a(b+c)
bc
 + 1 ,
b(a+c)
ca
 + 1 ,
c(a+b)
ab
 + 1are in A.P. [adding 1 to each term]
⇒
ab+ca+bc
bc
 ,
ab+bc+ca
ca
 ,
ca+bc+ab
ab
 are in A.P.
⇒
1
bc
,
1
ca
,
1
ab
 are in A.P. [Dividing by ab + bc + ca]
⇒ a, b, c are in A.P. [multiplying by abc]
Hence proved.
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