Sequences And Series Ques 18
- If $x$ be is the arithmetic mean and $y, z$ be two geometric means between any two positive numbers, then $\frac{y^3+z^3}{x y z}=\ldots$
(1997C, 2M)
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Answer:
Correct Answer: 18.$(2)$
Solution: Let the two positive numbers be $a$ and $b$.
$\therefore \quad x=\frac{a+b}{2} \quad \quad$ [since, $x$ is AM between $a$ and $b$ ]$\quad$ ……..(i)
and $\frac{a}{y}=\frac{y}{z}=\frac{z}{b}\quad$ [since, $y, z$ are GM’s between $a$ and $b$ ]
$\therefore \quad a=\frac{y^2}{z} \quad$ and $\quad b=\frac{z^2}{y}$
On substituting the values of $a$ and $b$ in Eq. (i), we get
$2 x=\frac{y^2}{z}+\frac{z^2}{y}$
$\Rightarrow \quad \frac{y^3+z^3}{y z}=2 x$
$\Rightarrow \quad \frac{y^3+z^3}{x y z}=2$
BETA
