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If a,b,c are in ap then show that1/ba, 1/ca, 1/ab are in ap

If a,b,c are in ap then show that1/ba, 1/ca, 1/ab are in ap

Grade:10

3 Answers

Vikas TU
14149 Points
7 years ago
If a,b,c are in ap, then
their common difference would be:
b – a and c-b
and therfore,
2b = a+c..................(1)
If 
1/ba, 1/ca, 1/cb are in A.P,
then
2/ca = 1/ba + 1/cb
taking L.C.M abc and calculating we get,
again
2b = a + c.................(2)
hence eqn. (1) and (2) are similar therefore this series is also in A.P.
mycroft holmes
272 Points
7 years ago
If a,b,c are in AP, then c,b,a are in AP which means
 
c/abc, b/abc, a/abc are in AP which gives 1/ab, 1/bc, 1ca are in AP.
ankit singh
askIITians Faculty 614 Points
3 years ago
swers are the finest of the finest.
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.
 

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