If a,b,c are in AP,
b-a = c-b
to prove 1/bc, 1/ca, 1/ab are in AP, we need to show 1/ca - 1/bc = 1/ab - 1/caLHS
1/ca - 1/bc = (b-a)/abc RHS:
1/ab - 1/ca = (c-b)/abc
but we know that (b-a) = (c-b)
thus LHS = RHS.Hence 1/bc, 1/ca, 1/ab are in AP.
Note: Calculation of 1/ca - 1/bc
You need to first take the LCM of ca and bc which is abc and do the calculation. I have calculated like this above.
Here instead of taking LCM, i have multiplied the terms, which is fine. You will get the same answer.