Algebra> If a,b,c are in ap then show that1/ba, 1/...

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

Vaibhav , 9 Years ago

Grade 10

3 Answers

Vikas TU

If a,b,c are in ap, then

their common difference would be:

b – a and c-b

and therfore,

2b = a+c..................(1)

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.

Last Activity: 9 Years ago

mycroft holmes

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.

Last Activity: 9 Years ago

ankit singh

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.

Last Activity: 5 Years ago