Question

Prove that a,b,c are in A.P. iff 1/bc,1/ca,1/ab are also in A.P.​

Answer 1

when 1/bc , 1/ca ,1/ab are in AP

1/ca - 1/bc = 1/ab - 1/ca

bc -ca/abc2 = ca - ab / a2 bc

c(b-a)/ c = a(c-b)/ a

then it becomes

b-a = c-b

i.e 2b = a+c

therefore a, b,c are in AP

Answer 2

Step-by-step explanation:

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.