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Question Number 56419 by gunawan last updated on 16/Mar/19

$$\mathrm{Let}a,b,c\mathrm{be}\mathrm{in}\mathrm{AP}\mathrm{and}\mid a\mid <1,\mid b\mid <1,\mid c\mid <1.\mathrm{If}$$$$x=1+a+a^2+...\mathrm{to}\mathrm{\infty }$$$$y=1+b+b^2+...\mathrm{to}\mathrm{\infty }$$$$z=1+c+c^2+...\mathrm{to}\mathrm{\infty }$$$$\mathrm{then}x,y,z\mathrm{are}\mathrm{in}$$

Answered by tanmay.chaudhury50@gmail.com last updated on 16/Mar/19

$$x=\frac{1}{1-a}y=\frac{1}{1-b}z=\frac{1}{1-c}$$$$1-a=\frac{1}{x}\to a=1-\frac{1}{x}$$$$b=1-\frac{1}{y}$$$$c=1-\frac{1}{z}$$$$2b=a+c$$$$2(1-\frac{1}{y})=1-\frac{1}{x}+1-\frac{1}{z}$$$$\frac{2}{y}=\frac{1}{x}+\frac{1}{z}sox,yandzinH.P$$

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