If $\frac{1}{x + y}, \frac{1}{2 y}, \frac{1}{y + z}$ are the three consecutive terms of an A.P. prove that x, y, z are the consecutive terms of a G.P.

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Solution

We have,

$\frac{1}{x + y}, \frac{1}{2 y}, \frac{1}{y + z}$ are the A.P.

We know that,

In terms of an A.P.

$T_{2} - T_{1} = T_{3} - T_{2}$

$\Rightarrow \frac{1}{2 y} - \frac{1}{x + y} = \frac{1}{y + z} - \frac{1}{2 y}$

$\Rightarrow \frac{x + y - 2 y}{2 y (x + y)} = \frac{2 y - y - z}{2 y (y + z)}$

$\Rightarrow \frac{x - y}{2 y (x + y)} = \frac{y - z}{2 y (y + z)}$

$\Rightarrow \frac{(x - y)}{(x + y)} = \frac{(y - z)}{(y + z)}$

$\Rightarrow (x - y) (y + z) = (x + y) (y - z)$

$\Rightarrow x y - y^{2} + x z - y z = x y + y^{2} - x z - y z$

$\Rightarrow x z - y z + x z + y z = 2 y^{2}$

$\Rightarrow 2 x z = 2 y^{2}$

$\Rightarrow y^{2} = x z$

Hence, $x, y, z$ in G.P.

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If 1x+y,12y,1y+z are the three consecutive terms of an A.P. prove that x, y, z are the consecutive terms of a G.P.

If $\frac{1}{x + y}, \frac{1}{2 y}, \frac{1}{y + z}$ are the three consecutive terms of an A.P. prove that x, y, z are the consecutive terms of a G.P.