Let a, b, c be in A.P. then 1/ab, 1/ca and 1/bc are in

Question

Question

Answer 1

Concept:

If a,b, c are in AP, then 2b = a + c or b - a = c - b

Calculation:

Given: a, b, c are in AP

Let 1/ab, 1/ca and 1/bc be in AP

⇒\(\rm \frac{1}{ca}\rm -\frac{1}{ab}=\rm \frac{1}{bc}-\rm \frac{1}{ca} \)

\(\rm \frac{1}{a}( \frac{1}{c}- \frac{1}{b})= \frac{1}{c}( \frac{1}{b}- \frac{1}{a})\\ ⇒ \frac{b-c}{abc}= \frac{a-b}{abc}\)

⇒ b - c = a - b

⇒2b = a + c

⇒ a,b, c are in AP, which is true.