Sequences and Series 4 Question 4
4. Three positive numbers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. Then, the common ratio of the GP is
(a) $\sqrt{2}+\sqrt{3}$
(b) $3+\sqrt{2}$
(c) $2-\sqrt{3}$
(d) $2+\sqrt{3}$
(2014 Main)
Show Answer
Solution:
- Let $a, a r, a r^{2}$ are in GP, where $(r>1)$.
On multiplying middle term by 2 , we have
$a, 2 a r, a r^{2}$ are in an AP.
$$ \begin{array}{rlrl} \Rightarrow & & 4 a r & =a+a r^{2} \\ \Rightarrow & r^{2}-4 r+1 & =0 \\ & \Rightarrow & r & =\frac{4 \pm \sqrt{16-4}}{2}=2 \pm \sqrt{3} \\ & \Rightarrow & r & =2+\sqrt{3} \quad \text { [since, AP is increasing] } \end{array} $$
