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  4. Let a, b, c be in AP. If 0 < a,b,c < 1 ,x=∑ limitsn=0∞ an, y=∑ limitsn=0∞ bn and z=∑ limitsn=0∞ cn, then

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Q. Let $a, b, c$ be in $AP$. If $ 0 < a,b,c < 1 ,x=\sum\limits_{n=0}^{\infty }{{{a}^{n}}}, $ $ y=\sum\limits_{n=0}^{\infty }{{{b}^{n}}} $ and $ z=\sum\limits_{n=0}^{\infty }{{{c}^{n}}}, $ then

KEAMKEAM 2008Sequences and Series

Solution:

Since, $ x=\sum\limits_{n=0}^{\infty }{{{a}^{n}}} $
$ \therefore $ $ x=1+a+{{a}^{2}}+....\infty $
$ \Rightarrow $ $ x=\frac{1}{1-a} $
$ \Rightarrow $ $ (1-a)x=1 $
$ \Rightarrow $ $ a=\frac{x-1}{x} $ Similarly, $ b=\frac{y-1}{y} $ and $ c=\frac{z-1}{z} $
Since, a, b and c are in AP.
$ \therefore $ $ b=\frac{a+c}{2} $
$ \Rightarrow $ $ \frac{y-1}{y}=\frac{\frac{x-1}{x}+\frac{z-1}{z}}{2} $
$ \Rightarrow $ $ 2xz(y-1)=y[z(x-1)+x(z-1)] $
$ \Rightarrow $ $ 2xyz-2xz=xyz-yz+xyz-xy $
$ \Rightarrow $ $ -2xz=-yz-xy $
$ \Rightarrow $ $ 2xz=xy+yz $