How many terms of the AP `-15, -13, -11, … ` are needed to make the sum `-55` ?

Answer

To find how many terms of the arithmetic progression (AP) `-15, -13, -11, ...` are needed to make the sum `-55`, we can follow these steps:

### Step 1: Identify the first term (A) and the common difference (D)
The first term \( A \) is the first term of the AP, which is:
\[ A = -15 \]

The common difference \( D \) can be calculated as:
\[ D = -13 - (-15) = -13 + 15 = 2 \]

### Step 2: Write the formula for the sum of the first \( n \) terms of an AP
The formula for the sum of the first \( n \) terms \( S_n \) of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]

### Step 3: Substitute the known values into the formula
We want to find \( n \) such that:
\[ S_n = -55 \]
Substituting \( A = -15 \) and \( D = 2 \) into the formula gives:
\[ -55 = \frac{n}{2} \times (2 \times -15 + (n - 1) \times 2) \]

### Step 4: Simplify the equation
Now we simplify the equation:
\[ -55 = \frac{n}{2} \times (-30 + 2n - 2) \]
\[ -55 = \frac{n}{2} \times (2n - 32) \]

### Step 5: Multiply both sides by 2 to eliminate the fraction
\[ -110 = n(2n - 32) \]

### Step 6: Rearrange the equation to form a quadratic equation
\[ 2n^2 - 32n + 110 = 0 \]

### Step 7: Simplify the quadratic equation
Dividing the entire equation by 2 gives:
\[ n^2 - 16n + 55 = 0 \]

### Step 8: Factor the quadratic equation
We need to factor this quadratic:
\[ n^2 - 11n - 5n + 55 = 0 \]
This can be factored as:
\[ (n - 11)(n - 5) = 0 \]

### Step 9: Solve for \( n \)
Setting each factor to zero gives:
1. \( n - 11 = 0 \) → \( n = 11 \)
2. \( n - 5 = 0 \) → \( n = 5 \)

### Conclusion
Thus, the number of terms needed to make the sum `-55` is either \( n = 5 \) or \( n = 11 \).