First Principles AI

Question 1

Find A.M. between the given numbers:

$2 + \sqrt{3}i,\ 2 - \sqrt{3}i$

$(a+b)^2,\ (a-b)^2$

Question 2

If 6, 11, 16 are three A.Ms. between $a$ and $b$ , find $a$ and $b$ .

Question 3

Insert five A.Ms. between $\sqrt{2}$ and $\dfrac{15}{\sqrt{2}}$ .

Question 4

The A.M. of two numbers is 7 and their product is 45. Find the numbers.

Question 5

If $n$ arithmetic means are inserted between $a$ and $b$ , prove that $d = \dfrac{b-a}{n+1}$ , where $d$ is the common difference.

Question 6

If $A$ is the A.M. between $a$ and $b$ , prove that $(a-A)^2 + (A-b)^2 = \dfrac{1}{2}(a-b)^2$ .

Question 7

For what value of $n$ , $\dfrac{a^{n+1} + b^{n+1}}{a^n + b^n}$ is the A.M. between $a$ and $b$ , where $a \neq b$ .

Questions