If n arithmetic means are inserted between two quantities a and b , then their sum is equal to :

Options

A. `n(a+b)`
B. `n/2 (a+b)`
C. `2n(a+b)`
D. `n/2 (a-b)`

Answer

To find the sum of n arithmetic means inserted between two quantities \( a \) and \( b \), we can follow these steps:

### Step 1: Understand the concept of arithmetic means
When we insert \( n \) arithmetic means between \( a \) and \( b \), we are essentially creating a sequence of \( n + 2 \) terms: \( a, M_1, M_2, \ldots, M_n, b \).

### Step 2: Determine the number of terms
The total number of terms in this sequence is \( n + 2 \) (including \( a \) and \( b \)).

### Step 3: Calculate the sum of the sequence
The sum of an arithmetic sequence can be calculated using the formula:
\[ \text{Sum} = \text{Number of terms} \times \text{Average of the first and last term} \]
In our case, the first term is \( a \) and the last term is \( b \).

### Step 4: Find the average of the first and last term
The average of the first and last term is:
\[ \text{Average} = \frac{a + b}{2} \]

### Step 5: Substitute into the sum formula
Now, substituting the number of terms and the average into the sum formula:
\[ \text{Sum} = (n + 2) \times \frac{a + b}{2} \]

### Step 6: Simplify the expression
This can be simplified to:
\[ \text{Sum} = \frac{(n + 2)(a + b)}{2} \]

### Conclusion
Thus, the sum of \( n \) arithmetic means inserted between \( a \) and \( b \) is:
\[ \text{Sum} = \frac{(n + 2)(a + b)}{2} \]