Question

The sum of three terms of a strictly increasing G.P is $\alpha S$ and the sum of the squares of these terms is $S^2$, then if $S=10\sqrt{3}$ then the greatest value of the middle term is,

• A. $5$
• B. $5\sqrt{3}$
• C. $10$
• D. $10\sqrt{3}$

Answer 1

The correct option is C $10$

Here, $S=10\sqrt{3}$

$\because a^2(\frac{1}{r^2}+1+r^2)=S^2$

or $\frac{1}{r^2}+1+r^2=\frac{S^2}{a^2}=\frac{300}{a^2}$

or $\frac{1}{r^2}+r^2=\frac{300}{a^2}-1$ (on transposition)

As $\frac{1}{r^2}+r^2>2$

$\Rightarrow \frac{300}{a^2}-1>2$

$\Rightarrow \frac{300}{a^2}>2+1=3$

$\Rightarrow 300>3a^2$

or $3a^2<300$

or $a^2<100$

or $a<\pm 10$

Thus, $-10<a<10$

$\Rightarrow$ Greatest value of $a$ (middle term) is $10$

Hence option $C$ is the answer.