Let $a,b,c$ be in AP. If $0

Question

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

Dr. Abubacker · Accepted Answer

(d)
Since, $x = 1 + a + a^{2} + ...\infty$
$\Rightarrow \ x = \frac{1}{1 - a}\ \ \  \Rightarrow \ \ a = \frac{x - 1}{x}$
Similarly, $b = \frac{y - 1}{y}\ \$and $\ \ c = \frac{z - 1}{z}$
Since, $a,b,c$ are in AP.
$	herefore b = \frac{a + c}{2}$
$\Rightarrow \frac{y - 1}{y} = \frac{\frac{x - 1}{x} + \frac{z - 1}{z}}{2}$
$\Rightarrow 2xz(y - 1) = y\lbrack z(x - 1) + x(z - 1)brack$
$\Rightarrow 2xz = xy + yz$

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

Explanation

Reviewed by Dr. Abubacker

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