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

$$\mathrm{If}\mathrm{a},\mathrm{b},\mathrm{c}\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}\mathrm{show}\mathrm{that}$$$$\frac{1}{\sqrt{\mathrm{b}}+\sqrt{\mathrm{c}}},\frac{1}{\sqrt{\mathrm{c}}+\sqrt{\mathrm{a}}},\frac{1}{\sqrt{\mathrm{b}}+\sqrt{\mathrm{a}}},\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}$$

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

$$a,b,c\mathrm{in}AP\Rightarrow b-a=c-b=d\left(\mathrm{common}\mathrm{diff}\right)$$$$\mathrm{If}{u}_1=\frac{1}{\sqrt{b}+\sqrt{c}},u_2=\frac{1}{\sqrt{c}+\sqrt{b}},u_3=\frac{1}{\sqrt{b}+\sqrt{a}}\mathrm{are}\mathrm{in}\mathrm{AP}$$$$\mathrm{then}{u}_2\mathrm{is}\mathrm{the}\mathrm{arithmetic}\mathrm{mean}\mathrm{of}{u}_1\mathrm{and}{u}_3$$$$\Rightarrow u_1+u_3=2u_2$$$$\frac{1}{\sqrt{b}+\sqrt{c}},\frac{1}{\sqrt{c}+\sqrt{a}},\frac{1}{\sqrt{b}+\sqrt{a}}\iff \frac{\sqrt{c}-\sqrt{b}}{c-b},\frac{\sqrt{c}-\sqrt{a}}{c-a},\frac{\sqrt{b}-\sqrt{a}}{b-a}$$$$\iff \frac{\sqrt{c}-\sqrt{b}}{d},\frac{\sqrt{c}-\sqrt{a}}{2d},\frac{\sqrt{b}-\sqrt{a}}{d}[c-a=(c-b)+(b-a)=2d]$$$$\mathrm{Now}{u}_1=\frac{\sqrt{c}-\sqrt{b}}{d},u_2=\frac{\sqrt{c}-\sqrt{a}}{2d},u_3=\frac{\sqrt{b}-\sqrt{a}}{d}$$$$u_1+u_3=\frac{\sqrt{c}-\sqrt{b}}{d}+\frac{\sqrt{b}-\sqrt{a}}{d}=\frac{\sqrt{c}-\sqrt{a}}{d}=2u_2$$

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