11th term of an A.P. is `13, 15 1/2, 18, 20 1/2, ` ……… is :

Options

A. 38
B. `40 1/2`
C. `43`
D. `45 1/2`

Answer

To find the 11th term of the given arithmetic progression (A.P.) which is: 13, 15 1/2, 18, 20 1/2, we will follow these steps:

### Step 1: Convert Mixed Fractions to Improper Fractions
First, we need to convert the mixed fractions into improper fractions or decimals for easier calculations.

1. **Convert 15 1/2**:
- \( 15 \frac{1}{2} = 15 + \frac{1}{2} = \frac{15 \times 2 + 1}{2} = \frac{30 + 1}{2} = \frac{31}{2} \)
- In decimal, \( \frac{31}{2} = 15.5 \)

2. **Convert 20 1/2**:
- \( 20 \frac{1}{2} = 20 + \frac{1}{2} = \frac{20 \times 2 + 1}{2} = \frac{40 + 1}{2} = \frac{41}{2} \)
- In decimal, \( \frac{41}{2} = 20.5 \)

Now, our A.P. can be rewritten as:
- 13, 15.5, 18, 20.5

### Step 2: Identify the First Term (a) and Common Difference (d)
- The first term \( a = 13 \)
- To find the common difference \( d \), we can subtract the first term from the second term:
- \( d = 15.5 - 13 = 2.5 \)

### Step 3: Use the Formula for the nth Term of an A.P.
The formula for the nth term of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]

Where:
- \( a_n \) is the nth term,
- \( a \) is the first term,
- \( n \) is the term number,
- \( d \) is the common difference.

### Step 4: Calculate the 11th Term
Now, we will calculate the 11th term using the formula:
- Here, \( n = 11 \), \( a = 13 \), and \( d = 2.5 \).

Substituting the values into the formula:
\[ a_{11} = 13 + (11 - 1) \cdot 2.5 \]
\[ a_{11} = 13 + 10 \cdot 2.5 \]
\[ a_{11} = 13 + 25 \]
\[ a_{11} = 38 \]

### Final Answer
The 11th term of the arithmetic progression is **38**.

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