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 as 5, 9, 13, ..., 185, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) of the AP is 5. The common difference \( d \) can be calculated as follows:
\[ d = 9 - 5 = 4 \]

### Step 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:
\[ A_n = a + (n - 1) \cdot d \]
In this case, we know that the last term \( A_n \) is 185.

### Step 3: Substitute the known values into the formula
Substituting the values of \( a \), \( d \), and \( A_n \) into the formula:
\[ 185 = 5 + (n - 1) \cdot 4 \]

### Step 4: Simplify the equation
First, subtract 5 from both sides:
\[ 185 - 5 = (n - 1) \cdot 4 \]
\[ 180 = (n - 1) \cdot 4 \]

### Step 5: Divide both sides by 4
Now, divide both sides by 4 to isolate \( n - 1 \):
\[ n - 1 = \frac{180}{4} \]
\[ n - 1 = 45 \]

### Step 6: Solve for n
Now, add 1 to both sides to find \( n \):
\[ n = 45 + 1 \]
\[ n = 46 \]

### Conclusion
The number of terms in the AP is 46.

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