How many terms of the A.P.,` -6,(-11)/(2),-5` ... are needed to give the sum-25 ?

Answer

To solve the problem of finding how many terms of the arithmetic progression (A.P.) -6, -11/2, -5, ... are needed to give the sum of -25, we can follow these steps:

### Step 1: Identify the first term (A) and the common difference (D)
The first term \( A \) is given as:
\[ A = -6 \]
To find the common difference \( D \), we calculate:
\[ D = A_2 - A_1 = -\frac{11}{2} - (-6) = -\frac{11}{2} + 6 = -\frac{11}{2} + \frac{12}{2} = \frac{1}{2} \]
Thus, the common difference \( D \) is:
\[ D = \frac{1}{2} \]

### Step 2: Use the formula for the sum of the first \( N \) terms of an A.P.
The formula for the sum of the first \( N \) terms \( S_N \) of an A.P. is:
\[ S_N = \frac{N}{2} \times (2A + (N - 1)D) \]
We know that \( S_N = -25 \), so we can set up the equation:
\[ -25 = \frac{N}{2} \times (2(-6) + (N - 1)\left(\frac{1}{2}\right)) \]

### Step 3: Simplify the equation
Substituting \( A \) and \( D \) into the equation gives:
\[ -25 = \frac{N}{2} \times (-12 + \frac{N - 1}{2}) \]
Multiplying both sides by 2 to eliminate the fraction:
\[ -50 = N \times (-12 + \frac{N - 1}{2}) \]
Distributing \( N \):
\[ -50 = -12N + \frac{N^2 - N}{2} \]
Now, multiply through by 2 to eliminate the fraction:
\[ -100 = -24N + N^2 - N \]
Rearranging gives:
\[ N^2 - 25N + 100 = 0 \]

### Step 4: Solve the quadratic equation
We can use the quadratic formula \( N = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -25, c = 100 \):
\[ N = \frac{25 \pm \sqrt{(-25)^2 - 4 \cdot 1 \cdot 100}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (-25)^2 - 400 = 625 - 400 = 225 \]
Thus:
\[ N = \frac{25 \pm 15}{2} \]
Calculating the two possible values:
\[ N_1 = \frac{40}{2} = 20 \quad \text{and} \quad N_2 = \frac{10}{2} = 5 \]

### Step 5: Verify which value of N satisfies the original sum condition
We need to check which of these values satisfies the condition \( S_N = -25 \):
1. For \( N = 5 \):
\[ S_5 = \frac{5}{2} \times (2(-6) + (5 - 1)\left(\frac{1}{2}\right)) = \frac{5}{2} \times (-12 + 2) = \frac{5}{2} \times (-10) = -25 \]
2. For \( N = 20 \):
\[ S_{20} = \frac{20}{2} \times (2(-6) + (20 - 1)\left(\frac{1}{2}\right)) = 10 \times (-12 + 9.5) = 10 \times -2.5 = -25 \]

Both values satisfy the condition, but since the question asks for how many terms are needed, we can conclude:
\[ \text{The number of terms needed is } N = 5. \]

### Final Answer:
The number of terms of the A.P. needed to give the sum of -25 is **5**.