How many terms of the A.P. : 24, 21, 18,…………... must be taken so that their sum is 78 ?

Answer

To solve the problem, we need to find how many terms of the arithmetic progression (A.P.) 24, 21, 18, ... must be taken so that their sum is 78.

### Step-by-Step Solution:

1. **Identify the first term (a) and the common difference (d)**:
- The first term \( a \) is 24.
- The second term is 21, so the common difference \( d \) can be calculated as:
\[ d = 21 - 24 = -3 \]

2. **Use the formula for the sum of the first n terms of an A.P.**:
- The formula for the sum \( S_n \) of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \]
- We know \( S_n = 78 \), \( a = 24 \), and \( d = -3 \). Substituting these values into the formula gives:
\[ 78 = \frac{n}{2} \left(2 \cdot 24 + (n-1)(-3)\right) \]

3. **Simplify the equation**:
- First, simplify the expression inside the parentheses:
\[ 2 \cdot 24 = 48 \]
\[ (n-1)(-3) = -3n + 3 \]
- So, the equation becomes:
\[ 78 = \frac{n}{2} \left(48 - 3n + 3\right) \]
\[ 78 = \frac{n}{2} \left(51 - 3n\right) \]

4. **Multiply both sides by 2 to eliminate the fraction**:
\[ 156 = n(51 - 3n) \]

5. **Rearrange the equation**:
\[ 156 = 51n - 3n^2 \]
\[ 3n^2 - 51n + 156 = 0 \]

6. **Divide the entire equation by 3 to simplify**:
\[ n^2 - 17n + 52 = 0 \]

7. **Factor the quadratic equation**:
- We need to find two numbers that multiply to 52 and add up to -17. The factors are -13 and -4.
\[ (n - 13)(n - 4) = 0 \]

8. **Solve for n**:
- Setting each factor to zero gives:
\[ n - 13 = 0 \quad \Rightarrow \quad n = 13 \]
\[ n - 4 = 0 \quad \Rightarrow \quad n = 4 \]

9. **Conclusion**:
- The number of terms that can be taken from the A.P. to achieve a sum of 78 is either 4 or 13.