Question

if a²,b²,c² are in A.P, then prove that 1/b+c, 1/c+a, 1/a+b are in A.P

Answer 1

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
who is in A.p.

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