If `f_(n) = {{:( (f_(n-1))/(2) " when " f_n-1 "is an even number"), (3* f_(n-1) + 1 " when " f_(n-1) " is an odd number "):}` and `f_(1) = 3` , then `f_(5) = `

Options

A. 1
B. 2
C. 4
D. 8

Explanation

To solve the problem step by step, we will use the recursive function defined in the question. Let's denote the function as \( f(n) \) for clarity.

Given:
- \( f(n) = \frac{f(n-1)}{2} \) when \( f(n-1) \) is even
- \( f(n) = 3 \cdot f(n-1) + 1 \) when \( f(n-1) \) is odd
- \( f(1) = 3 \)

We need to find \( f(5) \).

### Step 1: Calculate \( f(2) \)
Since \( f(1) = 3 \) (which is odd), we use the odd condition:
\[ f(2) = 3 \cdot f(1) + 1 = 3 \cdot 3 + 1 = 9 + 1 = 10 \]

### Step 2: Calculate \( f(3) \)
Now, \( f(2) = 10 \) (which is even), so we use the even condition:
\[ f(3) = \frac{f(2)}{2} = \frac{10}{2} = 5 \]

### Step 3: Calculate \( f(4) \)
Next, \( f(3) = 5 \) (which is odd), so we use the odd condition again:
\[ f(4) = 3 \cdot f(3) + 1 = 3 \cdot 5 + 1 = 15 + 1 = 16 \]

### Step 4: Calculate \( f(5) \)
Finally, \( f(4) = 16 \) (which is even), so we use the even condition:
\[ f(5) = \frac{f(4)}{2} = \frac{16}{2} = 8 \]

Thus, the final answer is:
\[ f(5) = 8 \]

### Summary of Steps:
1. Calculate \( f(2) \) using the odd condition from \( f(1) \).
2. Calculate \( f(3) \) using the even condition from \( f(2) \).
3. Calculate \( f(4) \) using the odd condition from \( f(3) \).
4. Calculate \( f(5) \) using the even condition from \( f(4) \).