An AP starts with a positive fraction and every alternate term is an integer. If the sum of the first 11 terms is 33, then find the fourth term.

Options

A. 2
B. 3
C. 5
D. 6

Answer

To solve the problem step by step, we will follow the logical deductions based on the information provided.

### Step 1: Understanding the Problem
We are given that:
- The first term \( A \) of an arithmetic progression (AP) is a positive fraction.
- Every alternate term is an integer.
- The sum of the first 11 terms is 33.

### Step 2: Using the Sum Formula for AP
The formula for the sum of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2A + (n-1)D) \]
For our case, \( n = 11 \) and \( S_{11} = 33 \):
\[ S_{11} = \frac{11}{2} \times (2A + 10D) = 33 \]

### Step 3: Simplifying the Equation
Now, we can simplify the equation:
\[ \frac{11}{2} \times (2A + 10D) = 33 \]
Multiplying both sides by 2:
\[ 11(2A + 10D) = 66 \]
Dividing both sides by 11:
\[ 2A + 10D = 6 \]

### Step 4: Rearranging the Equation
We can rearrange this equation to express it in terms of \( A \):
\[ 2A = 6 - 10D \]
\[ A = 3 - 5D \]

### Step 5: Analyzing the Terms
Since \( A \) is a positive fraction and every alternate term is an integer, we can deduce:
- The second term \( A + D \) must be an integer.
- The fourth term \( A + 3D \) must also be an integer.

### Step 6: Setting Up Conditions
From the above, we can conclude:
1. \( A \) is a fraction.
2. \( D \) must be such that \( A + D \) is an integer.

Let’s assume \( A = \frac{1}{2} \) (a simple positive fraction) and check if it satisfies the conditions:
\[ A = \frac{1}{2} \Rightarrow D = 3 - 5 \times \frac{1}{2} = 3 - \frac{5}{2} = \frac{1}{2} \]

### Step 7: Verifying the Terms
Now we can find the terms:
- First term \( A = \frac{1}{2} \)
- Second term \( A + D = \frac{1}{2} + \frac{1}{2} = 1 \) (integer)
- Third term \( A + 2D = \frac{1}{2} + 2 \times \frac{1}{2} = \frac{3}{2} \) (fraction)
- Fourth term \( A + 3D = \frac{1}{2} + 3 \times \frac{1}{2} = 2 \) (integer)

### Step 8: Finding the Fourth Term
The fourth term is:
\[ A + 3D = 2 \]

### Conclusion
The fourth term of the AP is \( 2 \).