Question

Which of the following is “Mathematical Average”?
(a) Arithmetic Mean
(b) Geometric Mean
(c) Harmonic mean
(d) All of the above.

Answer 1

Hint: We start solving the problem by recalling the definitions and formulas of given Mathematical Average, Arithmetic Mean, Geometric Mean, Harmonic Mean taking n terms. We then compare each definition and formula for the n terms we just defined to get the required answer.

Complete step by step answer:
According to the problem, we need to check which of the given options is equivalent to the Mathematical Average.
Let us recall the definitions of Mathematical Average, Arithmetic Mean, Geometric Mean and Harmonic Mean.
Mathematical Average: We know that the mathematical average is defined as the sum of the terms divided with the total number of terms that were being averaged.
i.e., sum of the terms ${{a}_{1}}$, ${{a}_{2}}$,……, ${{a}_{n}}$ is $Average=\dfrac{{{a}_{1}}+{{a}_{2}}+......+{{a}_{n}}}{n}$ ---(1).
Arithmetic Mean (A.M): If the numbers ${{b}_{1}}$, ${{b}_{2}}$,……, ${{b}_{n}}$ follows the arithmetic progression, then the arithmetic mean of these numbers is defined as $A.M=\dfrac{{{b}_{1}}+{{b}_{2}}+......+{{b}_{n}}}{n}$ ---(2).
Geometric Mean (G.M): If the numbers ${{c}_{1}}$, ${{c}_{2}}$,……, ${{c}_{n}}$ follows the geometric progression, then the geometric mean of these numbers is defined as $G.M=\sqrt[n]{{{c}_{1}}\times {{c}_{2}}\times ......\times {{c}_{n}}}$ ---(3).
Harmonic Mean (H.M): If the numbers ${{d}_{1}}$, ${{d}_{2}}$,……, ${{d}_{n}}$ follows the harmonic progression, then the harmonic mean of these numbers is defined as $H.M=\dfrac{n}{\dfrac{1}{{{d}_{1}}}+\dfrac{1}{{{d}_{2}}}+......+\dfrac{1}{{{d}_{n}}}}$ ---(4).
From equations (1), (2), (3) and (4), we can see that the mathematical average is equal to the arithmetic mean.
So, we can say that the Mathematical average is equal to the arithmetic mean.

So, the correct answer is “Option A”.

Note: We should know that the inverse of the terms present in the Harmonic progression lies in the Arithmetic progression. If we don’t know the formula of the Harmonic Mean, we can convert it to the arithmetic progression and use the Arithmetic mean to get the harmonic mean. Here we have used the daily used formula for the average for the mathematical average. We should know that there are different types of averages and those definitions should not be used unless they have been mentioned in the problem.