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2014-10-29T11:31:25+05:30

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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 
 \frac{1}{ca} - \frac{1}{bc} = \frac{bc-ca}{abc^{2}} = \frac{c(b-a))}{abc^{2}} = \frac{b-a}{abc}  
Here instead of taking LCM, i have multiplied the terms, which is fine. You will get the same answer.
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