Questions

1. Question 1

   Find G.M. between:

   (i)

   $-2$ and $8$

   (ii)

   $-2i$ and $8i$

   (iii)

   6 and 9

   SolutionTheory
2. Question 2

   Insert four real geometric means between 3 and 96.

   SolutionTheory
3. Question 3

   If both $x$ and $y$ are positive distinct real numbers, show that the geometric mean between $x$ and $y$ is less than their arithmetic mean.

   SolutionTheory
4. Question 4

   For what value of $n$, $\dfrac{a^n + b^n}{a^{n-1} + b^{n-1}}$ is the positive geometric mean between $a$ and $b$?

   SolutionTheory
5. Question 5

   The A.M. of two positive integral numbers exceeds their (positive) G.M. by 2 and their sum is 20, find the numbers.

   SolutionTheory
6. Question 6

   The A.M. between two numbers is 5 and their (positive) G.M. is 4. Find the numbers.

   SolutionTheory
7. Question 7

   The arithmetic mean between two positive numbers $a$ and $b$ is double their geometric mean. Prove that $a:b = 2+\sqrt{3}:2-\sqrt{3}$.

   SolutionTheory
8. Question 8

   If one geometric mean $G$ and two arithmetic means $p$ and $q$ are inserted between two positive numbers, show that $G^2 = (2p-q)(2q-p)$.

   SolutionTheory