Question

If a,b,c are in Ap . Then prove that ab+bc+ca are in Ap

Answer 1

We have to prove that bc, ca, and ab are in HP, which is another way of saying that 1/bc, 1/ca, and 1/ab are in AP. This simply means we have to prove that

2(1/ca) = 1/ab + 1/bc
=> 2/ca = 1/ab + 1/bc

We are given that a, b, and c are in AP, which means

2b = a + c

Diving the entire expression by abc, we'll have

=> 2b/abc = a/abc + c/abc
=> 2/ca = 1/bc + 1/ab
which means 1/bc, 1/ca, and 1/ab are in AP, or bc, ca and ab are in AP