R S Aggarwal Solutions for Exercise 6: EXERCISE 12F

Find two positive numbers $a$ and $b$, whose

$AM=25$ and $GM=20$

Find two positive numbers $a$ and $b$, whose

$AM=10$ and $GM=8$

Insert two geometric means between $9$ and $243$.

Insert three geometric means between $\frac{1}{3}$ and $432$.

Insert four geometric means between $6$ and $192$.

If $a, b, c$ are in $\mathrm{AP}, x$ is the $\mathrm{GM}$ between $a$ and $b; y$ is the $\mathrm{GM}$ between $b$ and $c;$ then show that $b^2$ is the $\mathrm{AM}$ between $x^2$ and $y^2$.

Show that the product of $n$ geometric means between $a$ and $b$ is equal to the $n^{th}$ power of the single $GM$ between $a$ and $b$.

If $AM$ and $GM$ of the roots of a quadratic equation are $10$ and $8$ respectively then obtain the quadratic equation.