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Algebra of Derivatives

If the 2nd , ...

Question

If the $2^{n d}, 5^{t h} a n d 9^{t h}$ terms of non-constant A.P. are in G.P., then the common ratio of this G.P. is:

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Solution

The correct option is A

$a + d, a + 4 d, a + 8 d a r e i n G . P . (a + 4 d)^{2} = (a + d) (a + 8 d) \Rightarrow a^{2} + 8 a d + 16 d^{2} = a^{2} + 9 a d + 8 d^{2} \Rightarrow 8 d^{2} = a d \Rightarrow \frac{a}{d} = 8 ∴ C o m m o n r a t i o = \frac{a + 4 d}{a + d} = \frac{8 + 4}{8 + 1} = \frac{4}{3}$

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