The numbers of terms of the series 54, 51, 48,…so that their sum is 513

Options

A. 18, 19
B. 16, 22
C. 25, 36
D. none

Answer

To solve the problem, we need to find the number of terms in the arithmetic progression (AP) series 54, 51, 48, ... such that their sum equals 513.

### Step-by-Step Solution:

1. **Identify the first term (a) and common difference (d)**:
- The first term \( a = 54 \).
- The common difference \( d = 51 - 54 = -3 \).

2. **Use the formula for the sum of the first n terms of an AP**:
- The formula for the sum \( S_n \) of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
- We know \( S_n = 513 \).

3. **Substitute the known values into the formula**:
- Plugging in the values of \( a \) and \( d \):
\[ 513 = \frac{n}{2} \times (2 \times 54 + (n - 1)(-3)) \]
- Simplifying this:
\[ 513 = \frac{n}{2} \times (108 - 3(n - 1)) \]
\[ 513 = \frac{n}{2} \times (108 - 3n + 3) \]
\[ 513 = \frac{n}{2} \times (111 - 3n) \]

4. **Multiply both sides by 2 to eliminate the fraction**:
\[ 1026 = n(111 - 3n) \]
\[ 1026 = 111n - 3n^2 \]

5. **Rearranging the equation**:
\[ 3n^2 - 111n + 1026 = 0 \]

6. **Dividing the entire equation by 3**:
\[ n^2 - 37n + 342 = 0 \]

7. **Using the quadratic formula to find n**:
- The quadratic formula is given by:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Here, \( a = 1, b = -37, c = 342 \):
\[ n = \frac{37 \pm \sqrt{(-37)^2 - 4 \cdot 1 \cdot 342}}{2 \cdot 1} \]
\[ n = \frac{37 \pm \sqrt{1369 - 1368}}{2} \]
\[ n = \frac{37 \pm 1}{2} \]

8. **Calculating the two possible values for n**:
- \( n = \frac{38}{2} = 19 \)
- \( n = \frac{36}{2} = 18 \)

9. **Verifying the values of n**:
- For \( n = 18 \):
\[ S_{18} = \frac{18}{2} \times (2 \times 54 + (18 - 1)(-3)) = 9 \times (108 - 51) = 9 \times 57 = 513 \]
- For \( n = 19 \):
\[ S_{19} = \frac{19}{2} \times (2 \times 54 + (19 - 1)(-3)) = \frac{19}{2} \times (108 - 54) = \frac{19}{2} \times 54 = 513 \]
- The 19th term is 0, thus it does not affect the sum.

### Conclusion:
Both \( n = 18 \) and \( n = 19 \) satisfy the condition, but since the 19th term does not contribute to the sum, we can conclude that the number of terms is either 18 or 19.