In an A.P., if `a = 8,a_(4), = 17, S_(n) = 148` then value of n is.

Options

A. 6
B. 8
C. 10
D. 12

Answer

To solve the problem step by step, we will follow the given information and apply the formulas related to Arithmetic Progressions (A.P.).

### Step-by-Step Solution

1. **Identify the Given Values**:
- First term \( a = 8 \)
- Fourth term \( a_4 = 17 \)
- Sum of the first \( n \) terms \( S_n = 148 \)

2. **Use the Formula for the \( n \)-th Term**:
The formula for the \( n \)-th term of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
For the fourth term \( a_4 \):
\[ a_4 = a + (4 - 1) \cdot d \]
Substituting the known values:
\[ 17 = 8 + 3d \]

3. **Solve for the Common Difference \( d \)**:
Rearranging the equation:
\[ 17 - 8 = 3d \]
\[ 9 = 3d \]
\[ d = \frac{9}{3} = 3 \]

4. **Use the Formula for the Sum of the First \( n \) Terms**:
The formula for the sum of the first \( n \) terms \( S_n \) is:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d) \]
Substituting the known values:
\[ 148 = \frac{n}{2} \cdot (2 \cdot 8 + (n - 1) \cdot 3) \]
Simplifying:
\[ 148 = \frac{n}{2} \cdot (16 + 3n - 3) \]
\[ 148 = \frac{n}{2} \cdot (3n + 13) \]

5. **Multiply Both Sides by 2 to Eliminate the Fraction**:
\[ 296 = n(3n + 13) \]
\[ 296 = 3n^2 + 13n \]

6. **Rearranging to Form a Quadratic Equation**:
\[ 3n^2 + 13n - 296 = 0 \]

7. **Use the Quadratic Formula**:
The quadratic formula is given by:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = 13 \), and \( c = -296 \).

8. **Calculate the Discriminant**:
\[ b^2 - 4ac = 13^2 - 4 \cdot 3 \cdot (-296) \]
\[ = 169 + 3552 = 3721 \]

9. **Substitute into the Quadratic Formula**:
\[ n = \frac{-13 \pm \sqrt{3721}}{6} \]
\[ n = \frac{-13 \pm 61}{6} \]

10. **Calculate the Two Possible Values for \( n \)**:
- First solution:
\[ n = \frac{48}{6} = 8 \]
- Second solution (negative root):
\[ n = \frac{-74}{6} \text{ (not valid since } n \text{ must be positive)} \]

11. **Final Answer**:
The value of \( n \) is \( 8 \).