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. (d)
Solution:
- 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,
$$ r=\frac{a+4 d}{a+d}=\frac{8 d+4 d}{8 d+d}=\frac{12 d}{9 d}=\frac{4}{3} $$
