If-a-b-c-are-in-A-P-show-that-1-b-c-1-c-a-1-b-a-are-in-A-P-

Question Number 177932 by Spillover last updated on 11/Oct/22

If a, b, c are in A . P show that

\frac{1}{\sqrt{b} + \sqrt{c}}, \frac{1}{\sqrt{c} + \sqrt{a}}, \frac{1}{\sqrt{b} + \sqrt{a}}, are in A . P

Answered by Ar Brandon last updated on 11/Oct/22

a, b, c in A P \Rightarrow b - a = c - b = d (common diff)

If u_{1} = \frac{1}{\sqrt{b} + \sqrt{c}}, u_{2} = \frac{1}{\sqrt{c} + \sqrt{b}}, u_{3} = \frac{1}{\sqrt{b} + \sqrt{a}} are in AP

then u_{2} is the arithmetic mean of u_{1} and u_{3}

\Rightarrow u_{1} + u_{3} = 2 u_{2}

\frac{1}{\sqrt{b} + \sqrt{c}}, \frac{1}{\sqrt{c} + \sqrt{a}}, \frac{1}{\sqrt{b} + \sqrt{a}} \Leftrightarrow \frac{\sqrt{c} - \sqrt{b}}{c - b}, \frac{\sqrt{c} - \sqrt{a}}{c - a}, \frac{\sqrt{b} - \sqrt{a}}{b - a}

\Leftrightarrow \frac{\sqrt{c} - \sqrt{b}}{d}, \frac{\sqrt{c} - \sqrt{a}}{2 d}, \frac{\sqrt{b} - \sqrt{a}}{d} [c - a = (c - b) + (b - a) = 2 d]

Now u_{1} = \frac{\sqrt{c} - \sqrt{b}}{d}, u_{2} = \frac{\sqrt{c} - \sqrt{a}}{2 d}, u_{3} = \frac{\sqrt{b} - \sqrt{a}}{d}

u_{1} + u_{3} = \frac{\sqrt{c} - \sqrt{b}}{d} + \frac{\sqrt{b} - \sqrt{a}}{d} = \frac{\sqrt{c} - \sqrt{a}}{d} = 2 u_{2}