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.

Explanation

As, these three values are in A.P.

`:. b(1/c+1/a) - a(1/b+1/c) = c(1/a+1/b) - b(1/c+1/a) `

`=>b/c+b/a-a/b-a/c = c/a+c/b-b/c-b/a`

`=>1/c(b-a)+((b^2-a^2)/(ab)) = 1/a(c-b)+((c^2-b^2)/(bc))`

`=>1/c(b-a)+((b-a)(b+a))/(ab)) = 1/a(c-b)+((c-b)(c+b))/(bc))`

`=>(b-a)(1/c+(b+a)/(ab)) = (c-b)(1/a+(c+b)/(cb))`

`=>(b-a)(1/c+1/a+1/b) = (c-b)(1/a+1/b+1/c)`

`=>b-a = c-b`

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