- Home
- Standard 11
- Mathematics
જો $x=\sum \limits_{n=0}^{\infty} a^{n}, y=\sum\limits_{n=0}^{\infty} b^{n}, z=\sum\limits_{n=0}^{\infty} c^{n}$, જ્યાં $a , b , c$ એ સમાંતર શ્રેણી$(A.P.)$ માં છે. $|a| < 1,|b| < 1,|c| < 1$, $abc$ $\neq 0$ તો:
$x, y, z$ એ $A.P.$ મા છે.
$\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ એ $A.P.$માં છે.
$x, y, z$ એ $G.P.$મા છે.
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1-(a+b+c)$
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 $A.P.$
