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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/ca LHS: 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.

Or

Here instead of taking LCM, i have multiplied the terms, which is fine. You will get the same answer.