If $a, b, c$ are in A.P and $a, x, b$ and $b, y, c$ are in G.P , show that $x^{2}, b^{2}, y^{2}$ are in A.P.

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Solution

We have,
$a, b, c$ are in A.P.
$2 b = a + c$ ……. (1)

$a, x, b$ are in G.P.
$x^{2} = a b$ …… (2)

$b, y, c$ are in G.P.
$y^{2} = b c$ ……. (3)

From equation (1) and (2), we get
$x^{2} = (2 b - c) b$
$x^{2} = 2 b^{2} - b c$

On putting the value of $b c$ in above expression, we get
$x^{2} = 2 b^{2} - y^{2}$
$2 b^{2} = x^{2} + y^{2}$

So, $x^{2}, b^{2}, y^{2}$ are in A.P.

Hence, proved.

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