The number of terms of an AP 5, 9, 13,…. 185 is

Options

A. 31
B. 51
C. 41
D. 40

Answer

To find the number of terms in the arithmetic progression (AP) given by 5, 9, 13, ..., 185, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (a) and the common difference (d)**:
- The first term \( a = 5 \).
- The second term \( a_2 = 9 \).
- The common difference \( d = a_2 - a_1 = 9 - 5 = 4 \).

2. **Use the formula for the nth term of an AP**:
- The formula for the nth term \( a_n \) of an AP is given by:
\[ a_n = a + (n - 1) \cdot d \]
- Here, we know the nth term \( a_n = 185 \).

3. **Set up the equation**:
- Substitute the known values into the formula:
\[ 185 = 5 + (n - 1) \cdot 4 \]

4. **Simplify the equation**:
- First, subtract 5 from both sides:
\[ 185 - 5 = (n - 1) \cdot 4 \]
\[ 180 = (n - 1) \cdot 4 \]
- Next, divide both sides by 4:
\[ \frac{180}{4} = n - 1 \]
\[ 45 = n - 1 \]

5. **Solve for n**:
- Add 1 to both sides:
\[ n = 45 + 1 = 46 \]

6. **Conclusion**:
- The number of terms in the AP is \( n = 46 \).

### Final Answer:
The number of terms of the AP is **46**.