Prove that the sum of `n` arithmetic means between two numbers in `n` times the single. A.M. between them.

Explanation

To prove that the sum of \( n \) arithmetic means between two numbers \( A \) and \( B \) is \( n \) times the single arithmetic mean between them, we will follow these steps:

### Step-by-Step Solution

1. **Define the Numbers and the Means**:
Let the two numbers be \( A \) and \( B \). We will insert \( n \) arithmetic means between them. The sequence will be:
\[ A, A_1, A_2, A_3, \ldots, A_n, B \]
where \( A_1, A_2, \ldots, A_n \) are the \( n \) arithmetic means.

2. **Identify the Common Difference**:
Since the sequence is an arithmetic progression (AP), let \( d \) be the common difference. The terms can be expressed as:
\[ A_1 = A + d, \quad A_2 = A + 2d, \quad A_3 = A + 3d, \quad \ldots, \quad A_n = A + nd \]

3. **Express the Last Term**:
The last term \( A_n \) can also be expressed in terms of \( A \) and \( B \):
\[ A_n = A + nd = B \]
From this, we can find the common difference \( d \):
\[ nd = B - A \quad \Rightarrow \quad d = \frac{B - A}{n} \]

4. **Calculate the Sum of the Arithmetic Means**:
The sum of the arithmetic means \( S \) is given by:
\[ S = A_1 + A_2 + A_3 + \ldots + A_n \]
Using the formula for the sum of an arithmetic series:
\[ S_n = \frac{n}{2} (A_1 + A_n) \]
Here, \( A_1 = A + d \) and \( A_n = B \):
\[ S = \frac{n}{2} \left((A + d) + B\right) \]

5. **Substitute for \( d \)**:
Substitute \( d = \frac{B - A}{n} \) into the sum:
\[ S = \frac{n}{2} \left(A + \frac{B - A}{n} + B\right) \]
Simplifying this gives:
\[ S = \frac{n}{2} \left(A + B + \frac{B - A}{n}\right) = \frac{n}{2} \left(\frac{nA + nB + B - A}{n}\right) \]
\[ S = \frac{n}{2} \left(\frac{(n + 1)A + (n - 1)B}{n}\right) \]

6. **Express in Terms of the Arithmetic Mean**:
The single arithmetic mean \( M \) between \( A \) and \( B \) is:
\[ M = \frac{A + B}{2} \]
Therefore, we can express the sum \( S \) as:
\[ S = n \cdot M \]

7. **Conclusion**:
Thus, we have shown that the sum of \( n \) arithmetic means between \( A \) and \( B \) is equal to \( n \) times the single arithmetic mean between \( A \) and \( B \):
\[ S = n \cdot \frac{A + B}{2} \]
Hence, the statement is proved.