What is the least possible sum of the A.P. -23,-19 , -15 , … :

Options

A. `-3`
B. `-78`
C. `-87`
D. none of these

Answer

To find the least possible sum of the arithmetic progression (A.P.) given by the terms -23, -19, -15, ..., we can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the A.P. is -23, and the common difference \( d \) can be calculated as:
\[ d = -19 - (-23) = -19 + 23 = 4 \]

### Step 2: Write the formula for the sum of the first \( n \) terms of an A.P.
The sum \( S_n \) of the first \( n \) terms of an A.P. is given by the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Substituting the values of \( a \) and \( d \):
\[ S_n = \frac{n}{2} \times (2(-23) + (n-1)(4)) \]

### Step 3: Simplify the sum formula
Now, simplify the expression:
\[ S_n = \frac{n}{2} \times (-46 + 4(n-1)) \]
\[ = \frac{n}{2} \times (-46 + 4n - 4) \]
\[ = \frac{n}{2} \times (4n - 50) \]
\[ = 2n^2 - 25n \]

### Step 4: Find the minimum sum
To find the minimum value of \( S_n \), we can differentiate \( S_n \) with respect to \( n \) and set the derivative to zero:
\[ \frac{dS_n}{dn} = 4n - 25 \]
Setting the derivative to zero:
\[ 4n - 25 = 0 \implies n = \frac{25}{4} = 6.25 \]

### Step 5: Check the second derivative
To confirm that this value of \( n \) gives a minimum, we check the second derivative:
\[ \frac{d^2S_n}{dn^2} = 4 \]
Since \( 4 > 0 \), the function \( S_n \) has a minimum at \( n = 6.25 \).

### Step 6: Calculate the minimum sum
Now substitute \( n = 6.25 \) back into the sum formula:
\[ S_{6.25} = 2(6.25)^2 - 25(6.25) \]
Calculating \( (6.25)^2 = 39.0625 \):
\[ S_{6.25} = 2 \times 39.0625 - 25 \times 6.25 \]
\[ = 78.125 - 156.25 \]
\[ = -78.125 \]

### Conclusion
Thus, the least possible sum of the A.P. is approximately:
\[ \boxed{-78.125} \]