Suppose the sequence $a_{1}, a_{2}, a_{3} . . . . .$ is an arithmetic progression of distinct numbers such that the $a_{1}, a_{2}, a_{4}, a_{6} . . . . .$ is a geometric progression. The common ration of the geometric progression is:

- 2

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4

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a_{1}

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not determinable

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Solution

The correct option is B $- 2$
$a_{1}, a_{2}, a_{3}, a_{4}, . . . .$ are in AP
$a_{4} = a_{1} + 3 d d = (a_{2} - a_{1})$
$a_{1}, a_{2}, a_{4}, a_{6}, . . . .$ are in GP
$a_{4} = a_{1} r^{3} r = (\frac{a_{2}}{a_{1}})$
$a_{1} r^{3} = a_{1} + 3 d r^{3} = [\frac{a_{1} + 3 d}{a_{1}}] = [\frac{a_{1} + 3 a_{2} - 3 a_{1}}{a_{1}}] = (\frac{3 a_{2} - 2 a_{1}}{a_{1}}) r^{3} = (3 r - 2) r^{3} - r^{2} + r^{2} - 3 r + 2 = 0 r^{2} (r - 1) + r^{2} - r - 2 r + 2 = 0 r^{2} (r - 1) + (r - 1) (r - 2) = 0 (r - 1) (r^{2} - r + 2) = 0 (r - 1) (r^{2} + 2 r - r - 2) = 0 (r - 1) (r - 1) (r + 2) = 0 \Rightarrow r \neq 1, r = - 2$

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Suppose the sequence a1,a2,a3..... is an arithmetic progression of distinct numbers such that the a1,a2,a4,a6.....is a geometric progression. The common ration of the geometric progression is:

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