Sequences and Series 3 Question 7

7.

If the $2$ nd, $5$ th and $9$ th terms of a non-constant $AP$ are in GP, then the common ratio of this GP is

(2016 Main)

(a) $\frac{8}{5}$

(b) $\frac{4}{3}$

(c) 1

(d) $\frac{7}{4}$

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Answer:

Correct Answer: 7. (b)

Solution:

  1. Let $a$ be the first term and $d$ be the common difference.

Then, we have $a+d, a+4 d, a+8 d$ in GP,

i.e. $\quad(a+4 d)^{2}=(a+d)(a+8 d)$

$\Rightarrow \quad a^{2}+16 d^{2}+8 a d=a^{2}+8 a d+a d+8 d^{2}$

$\Rightarrow \quad 8 d^{2}=a d$

$\Rightarrow \quad 8 d=a$

$[\because d \neq 0]$

Now, common ratio,

$ r=\frac{a+4 d}{a+d}=\frac{8 d+4 d}{8 d+d}=\frac{12 d}{9 d}=\frac{4}{3} $



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