If A is an arithmetic mean between two number | vedclass

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(A) Let the two numbers be $a$ and $b$. The arithmetic mean $A$ between $a$ and $b$ is given by $A = \frac{a+b}{2}$. Let $A_1, A_2, \dots, A_n$ be the

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$S = nA$

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(A) Let the two numbers be $a$ and $b$. The arithmetic mean $A$ between $a$ and $b$ is given by $A = \frac{a+b}{2}$. Let $A_1, A_2, \dots, A_n$ be the $n$ arithmetic means between $a$ and $b$. Then $a, A_1, A_2, \dots, A_n, b$ form an arithmetic progression with $n+2$ terms. The sum of these $n$ arithmetic means is $S = A_1 + A_2 + \dots + A_n$. In an arithmetic progression,the sum of $n$ arithmetic means inserted between $a$ and $b$ is equal to $n$ times the single arithmetic mean between $a$ and $b$. Mathematically,$S = \sum_{i=1}^{n} A_i = n \left( \frac{a+b}{2} \right) = nA$. Therefore,$S = nA$.

If $A$ is an arithmetic mean between two numbers and $S$ is the sum of $n$ arithmetic means between the same two numbers,then:

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