Sequences and Series 5 Question 22
13. The sum of the squares of three distinct real numbers, which are in GP, is $S^{2}$. If their sum is $a S$, then show that
$$ a^{2} \in \frac{1}{3}, 1 \cup(1,3) $$
$(1986,5 M)$
Integer Answer Type Questions
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Solution:
- Here, $\alpha \in\left(0, \frac{\pi}{2}\right) \Rightarrow \tan \alpha>0$
$$ \therefore \quad \frac{\sqrt{x^{2}+x}+\frac{\tan ^{2} \alpha}{\sqrt{x^{2}+x}}}{2} \geq \sqrt{\sqrt{x^{2}+x} \cdot \frac{\tan ^{2} \alpha}{\sqrt{x^{2}+x}}} $$
[using AM $\geq GM$ ]
$\Rightarrow \quad \sqrt{x^{2}+x}+\frac{\tan ^{2} \alpha}{\sqrt{x^{2}+x}} \geq 2 \tan \alpha$
