Question

If the arithmetic and geometric mean of ‘a’ and ‘b’ be A and G respectively, then the value of \[A-G\] will be
1) \[\dfrac{\left( a-b \right)}{2}\]
2) \[\dfrac{\left( a+b \right)}{2}\]
3) \[{{\left[ \dfrac{\left( \sqrt{a}-\sqrt{b} \right)}{2} \right]}^{2}}\]
4) \[\dfrac{2ab}{a+b}\]

Answer 1

Hint: We are given the arithmetic mean and the geometric mean of two numbers ‘a’ and ‘b ‘as A and G respectively. We will first expand the expressions for A and G in terms of the given numbers ‘a’ and ‘b’. Then, we will have the expressions as \[A=\dfrac{a+b}{2}\] and \[G=\sqrt{ab}\]. Then, we will evaluate the value after substituting these expressions in \[A-G\]. On solving, we will have the required value of the expression.

Complete step by step solution:
According to the given question, we are given an arithmetic mean and geometric mean of two numbers ‘a’ and ‘b’. We are asked to find the value of the expression \[A-G\].
Arithmetic mean is the ratio of the sum of the numbers in a collection and count of the numbers in the collection.
Geometric mean indicates the central tendency of a collection of numbers using the product of their values.
Given that, arithmetic mean of ‘a’ and ‘b’ is \[A\]
We can expand and write it as,
\[A=\dfrac{a+b}{2}\] ---(1)
Similarly, we are given that the geometric mean of ‘a’ and ‘b’ is \[G\], which we can write it as,
\[G=\sqrt{ab}\] ---(2)
Now, we will substitute the values of equation (1) and (2) in the expression,
\[A-G\]
\[\begin{align}
  & \Rightarrow \dfrac{a+b}{2}-\sqrt{ab} \\
 & \Rightarrow \dfrac{a+b}{2}-\dfrac{2\sqrt{ab}}{2} \\
 & \Rightarrow \dfrac{a+b-2\sqrt{ab}}{2} \\
\end{align}\]
Simplifying the expression further. As we know that, \[{{\left( \sqrt{a}-\sqrt{b} \right)}^{2}}=a+b-2\sqrt{ab}\]
so we have the expression as,
\[\Rightarrow \dfrac{{{\left( \sqrt{a}-\sqrt{b} \right)}^{2}}}{2}\]
\[\Rightarrow {{\left[ \dfrac{\left( \sqrt{a}-\sqrt{b} \right)}{\sqrt{2}} \right]}^{2}}\]
Therefore, the correct option is 3) \[{{\left[ \dfrac{\left( \sqrt{a}-\sqrt{b} \right)}{2} \right]}^{2}}\].

Note: The two types of mean given should not be confused. The equation (1) of arithmetic mean should be written correctly. Also, the equation (2) of geometric mean should be written clearly and with correct powers. While substituting the values in the given expression, the values should be put in the correct order to avoid any errors.