If a, b, c are in A.P., prove that
1/√b+√c, 1/√c+√a, 1/√a+√b are in A.P.
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Answer:
Let
1/b+c, 1/c+a, 1/a+b
multiply (a+b+c)
then
1+a/b+c, 1+b/c+a, 1+c/a+b
add -1 in series
now
a/b+c, b/c+a, c/a+b
again multiply (a+b+c)
now
a + a^2/b+c, b + b^2/a+c , c + c^2/a+b
now add -(ab+ac)
you get
a^2 , b^2 , c^2
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