Prove that a,b,c are in A.P. iff 1/bc,1/ca,1/ab are also in A.P.

Consider 1/bc, 1/ca, 1/ab are in AP,

So, 1/ca = (1/bc + 1/ab)/2

or, 1/ca = (a+c)/2abc

or, 1/~~ca~~ = (a+c)/2~~a~~b~~c~~

or, 1 = (a+c)/2b

or, b = (a+c)/2

Thus, 'b' is A.M. of 'a' and 'c',

∴ a, b, c are in A.P. if 1/bc, 1/ca, 1/ab are in A.P.

**Proved**