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8. Sequences and Series
medium
If $x=\sum \limits_{n=0}^{\infty} a^{n}, y=\sum\limits_{n=0}^{\infty} b^{n}, z=\sum\limits_{n=0}^{\infty} c^{n}$, where $a , b , c$ are in $A.P.$ and $|a| < 1,|b| < 1,|c| < 1$, $abc \neq 0$, then
A
$x, y, z$ are in $A.P.$
B
$\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in $A.P.$
C
$x, y, z$ are in $G.P.$
D
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1-(a+b+c)$
(JEE MAIN-2022)
Solution
$x =1+ a + a ^{2}=\ldots \ldots \ldots .$
$x=\frac{1}{1-a} \Rightarrow a=1-\frac{1}{x}$
$y=\frac{1}{1-b} \Rightarrow b=1-\frac{1}{y}$
$z=\frac{1}{1-c} \Rightarrow c=1-\frac{1}{z}$
$a , b , c$ are in $A.P.$
$\Rightarrow 1-\frac{1}{x}, 1-\frac{1}{y}, 1-\frac{1}{z}$ are in $A.P.$
$\Rightarrow-\frac{1}{x},-\frac{1}{y},-\frac{1}{z}$ are in $A.P.$
$\Rightarrow \frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in $A.P.$
Standard 11
Mathematics
