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a, b, c are in Geometric progression, a, p, q are in Arithmetic progression, 2a,b + p,c + q are in Geometric progression, then common difference of A.P has the possible values

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a

$(a - b) (1 \pm \sqrt{2})$

b

$(ab) (1 \pm \sqrt{2})$

c

$(a - b) (3 \pm \sqrt{2})$

d

$(a - b) (1 \pm \sqrt{3})$

answer is A.

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Detailed Solution

$b^{2} = ac : 2 p = a + q : (b + p)^{2} = 2 a (c + q)$
Let p = a + d;q = a + 2d
Then $(b + a + d)^{2} = 2 a (\frac{b^{2}}{a} + a + 2 d)$
$\begin{array}{l} d^{2} - 2 ad + 2 bd = a^{2} + b^{2} - 2 ab \\ = (a - b) \pm \sqrt{2} (a - b) = (a - b) (1 \pm \sqrt{2}) \end{array}$

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a, b, c are in Geometric progression, a, p, q are in Arithmetic progression, 2a,b + p,c + q are in Geometric progression, then common difference of A.P has the possible values