Question

If $A$ be an arithmetic mean between two numbers and $S$ be the sum of n arithmetic means between the same numbers, then
$1)S = nA$
$2)A = nS$
$3)A = S$
$4)$None of these

Answer 1

First, we will see about the Arithmetic mean that is mentioned in the given problem.
The arithmetic mean is given by the sum of the observations divided by the number of observations.
Then we need to find the required sum of the given sequence and the number of terms in it to get the required answer.
By using the notation, $S,A$ we can find the relation.
Formula used:
Arithmetic mean can be expressed as $A = \dfrac{S}{n}$, where $A$ is the arithmetic mean, $S$ is the sum of the given sequence observations and n is the number of given sequence observations.
The sum of the Arithmetic mean can be expressed as $S = \dfrac{n}{2}(a + l)$where l is the last term.

Complete step by step answer:
From the given that $A$, be the arithmetic mean between two numbers and let it be $a$ and $b$.
Thus, by the use of the arithmetic mean formula we get, $A = \dfrac{S}{n}$
Here $S$ is the sum of the given two terms and n is the total count.
Thus, we get, $A = \dfrac{S}{n} \Rightarrow \dfrac{{a + b}}{2}$which is the AM of two numbers.
Now, $S$ be the sum of n arithmetic means between the same numbers where the formula for the sum of the Arithmetic mean can be expressed as $S = \dfrac{n}{2}(a + l)$
But we need to apply the same numbers which we used for AM, then we get $S = \dfrac{n}{2}(a + b)$( $a$ is the first number and $b$ is the last number)
Hence comparing both equations we get, $S = \dfrac{n}{2}(a + b) = n\dfrac{{(a + b)}}{2} = nA$ where $A = \dfrac{{a + b}}{2}$

So, the correct answer is “Option 1”.

Note: Arithmetic mean is the average or mean of the given set of numbers which is computed by adding all the terms in the set of numbers and dividing the sum by the total number of terms in the given.
The geometric mean is the mean value or the central term in the set of numbers in the geometric progression. Geometric means of sequence with the n terms is computed as the nth root of the product of all the terms in the sequence taken.
The harmonic mean is one of the types of determining the average. It is computed by dividing the number of values in the sequence by the sum of reciprocals of the terms.