If the middle term amongst any odd number (n) consecutive terms of an A.P, is m, then their sum is (a) `2m^2n` (b)`(mn)/2` (c)`mn` (d)`mn^2`

Options

A. 2mmn
B. mn/2
C. mn
D. `mn^(2)`

Answer

To solve the problem, we need to find the sum of n consecutive terms of an arithmetic progression (A.P.) when the middle term is given as m. Let's go through the solution step by step.

### Step 1: Understanding the A.P. Terms
Let the first term of the A.P. be \( a \) and the common difference be \( d \). The n consecutive terms can be represented as:
\[ a, \, a + d, \, a + 2d, \, \ldots, \, a + (n-1)d \]

### Step 2: Identifying the Middle Term
Since \( n \) is an odd number, the middle term can be found at the position \( \frac{n + 1}{2} \). Thus, the middle term \( m \) can be expressed as:
\[ m = a + \left(\frac{n - 1}{2}\right)d \]

### Step 3: Finding the Sum of n Terms
The sum \( S_n \) of the first \( n \) terms of an A.P. is given by the formula:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

### Step 4: Substitute for \( a \) and \( d \)
From the equation of the middle term, we can express \( a \) in terms of \( m \):
\[ a = m - \left(\frac{n - 1}{2}\right)d \]

Now substitute \( a \) back into the sum formula:
\[ S_n = \frac{n}{2} \times \left(2\left(m - \left(\frac{n - 1}{2}\right)d\right) + (n - 1)d\right) \]

### Step 5: Simplifying the Expression
Now, simplify the expression inside the parentheses:
\[ S_n = \frac{n}{2} \times \left(2m - (n - 1)d + (n - 1)d\right) \]
\[ S_n = \frac{n}{2} \times 2m \]
\[ S_n = n \times m \]

### Conclusion
Thus, the sum of the n consecutive terms of the A.P. when the middle term is \( m \) is:
\[ S_n = mn \]

### Final Answer
The correct option is (c) \( mn \).

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