Question

Arithmetic Mean and Geometric Mean of two positive real numbers a and b are equal if and only if *

a=b

a<b

a>b

None of these

​

Answer 1

Arithmetic Mean and Geometric Mean of two positive real numbers a and b are equal if and only if $a=b$.

Explanation:

• Arithmetic mean, $=\frac{a+b}{2}$ and Geometric mean, $=\sqrt{ab}$.
• When $a=b$, we get $\frac{a+b}{2}=\frac{a+a}{2}=\frac{2a}{2}=a$.
• And, $\sqrt{ab}=\sqrt{a^{2} } =a$.

Answer 2

The Arithmetic Mean and Geometric Mean of two positive real numbers a and b are equal if and only if a=b.

• When two numbers are equal, the values of their AM and GM will also be equal. Here, a=b.
• So, the AM of a and b will be (a+b)/2=(a+a)/2=2a/2=a.
• Similarly, the GM of a and b will be $\sqrt{ab}$=$\sqrt{a^{2} }$=a.
• But if the values of a or b are greater or less than the other, their arithmetic and geometric mean would be different.
• So, a=b is the right option.