`{:("Column A" , "The functons f is defined for all","ColumnB"),(, f(n) = n/(n+1),),( f(1)xx f(2), ,f(2) xx f(3)):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to evaluate the functions given in Column A and Column B and compare their values.

### Step-by-Step Solution:

1. **Define the function**:
The function \( f(n) \) is defined as:
\[ f(n) = \frac{n}{n + 1} \]

2. **Calculate \( f(1) \)**:
\[ f(1) = \frac{1}{1 + 1} = \frac{1}{2} \]

3. **Calculate \( f(2) \)**:
\[ f(2) = \frac{2}{2 + 1} = \frac{2}{3} \]

4. **Calculate \( f(3) \)**:
\[ f(3) = \frac{3}{3 + 1} = \frac{3}{4} \]

5. **Calculate Column A**:
Column A is defined as \( f(1) \times f(2) \):
\[ \text{Column A} = f(1) \times f(2) = \frac{1}{2} \times \frac{2}{3} = \frac{1 \times 2}{2 \times 3} = \frac{2}{6} = \frac{1}{3} \]

6. **Calculate Column B**:
Column B is defined as \( f(2) \times f(3) \):
\[ \text{Column B} = f(2) \times f(3) = \frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2} \]

7. **Compare Column A and Column B**:
- Column A = \( \frac{1}{3} \) (approximately 0.333)
- Column B = \( \frac{1}{2} \) (approximately 0.5)

Since \( \frac{1}{3} < \frac{1}{2} \), we conclude that:
\[ \text{Column B is greater than Column A} \]

### Final Answer:
The value of Column B is greater than the value of Column A.

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