R D Sharma Solutions for Chapter: Geometric Progressions, Exercise 6: EXERCISE 20.6

Find the geometric means of the following pair of numbers:
 $2$ and $8$

Find the geometric means of the following pairs of numbers:
 $-8$ and $-2$

The sum of two numbers is $6$ times their geometric means, show that the numbers are in the ratio $\left(3+2\sqrt{2}\right):(3-2\sqrt{2})$.

If AM and GM of roots of a quadratic equation are $8$ and $5$ respectively, then obtain the quadratic equation.

If AM and GM of two positive numbers $a$ and $b$ are $10$ and $8$ respectively, find the numbers

Prove that the product of $n$ geometric means between two quantities is equal to the $n^{th}$ power of a geometric mean of those two quantities.

If the A.M. of two positive numbers $a$ and $b\left(a>b\right)$ is twice their geometric mean. Prove that: $a:b=\left(2+\sqrt{3}\right):\left(2-\sqrt{3}\right)$.

If one A.M., $A$ and two geometric means $G_1$ and $G_2$ inserted between any two positive numbers, show that $\frac{G_1^2}{G_2}+\frac{G_2^2}{G_1}=2A$