Question

if a,b,c are in ap then proove that 1/ab, 1/bc, 1/ca​

Answer 1

Answer:

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 

=1/ca-1/bc

=(bc-ca) /abc²

=c(b-a) / abc²

=b-a/abc

 

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

Hence Proved

Step-by-step explanation:

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