Question

If the arithmetic mean of two numbers be $A$ and the geometric mean be $G$, then the numbers will be:

• A. $A\pm \left(A^2-G^2\right)$
• B. $\sqrt{A}\pm \sqrt{\left(A^2-G^2\right)}$
• C. $A\pm \sqrt{(A+G)(A-G)}$
• D. $\frac{(A\pm \sqrt{(A+G)(A-G))}}{2}$

Answer 1

The correct option is C

$A\pm \sqrt{(A+G)(A-G)}$

Explanation for the correct option.

Step 1: Arithmetic & geometric mean

Let the arithmetic & geometric mean be $A\&G$ respectively.

Let the number be $x,y$.
$\begin{array}{rcl}\frac{\mathrm{x}+\mathrm{y}}{2}& =& \mathrm{A}\dots \dots ..\left(1\right)\\ \sqrt{\mathrm{xy}}& =& \mathrm{G}\\ & \Rightarrow & \mathrm{y}=\frac{{\mathrm{G}}^2}{\mathrm{x}}\end{array}$

Step 2: Solve for $x$
Put the value of $y$ in equation (1) we have:
$\begin{array}{rcl}\frac{x+\frac{G^2}{x}}{x}& =& A\\ x^2+G^2& =& 2Ax\\ x^2-2Ax+G^2& =& 0\\ x& =& \frac{2A\pm \sqrt{4A^2-4G^2}}{2}\\ x& =& A\pm \sqrt{A^2-G^2}\\ x& =& A\pm \sqrt{\left(A+G\right)\left(A-G\right)}\end{array}$
Hence, option C is correct.