The set {x : x is a positive integer less than 6 and `3^(x)-1` is an even number} in roster form is

Options

A. `{1,2,3,4,5}`
B. `{1,2,3,4,5,6}`
C. `{2,4,6}`
D. `{1,3,5}`

Answer

To solve the problem, we need to find the set of positive integers \( x \) that are less than 6 and for which \( 3^x - 1 \) is an even number. Let's break it down step by step.

### Step 1: Identify the range of \( x \)
Since \( x \) is a positive integer less than 6, the possible values for \( x \) are:
\[ x = 1, 2, 3, 4, 5 \]

### Step 2: Evaluate \( 3^x - 1 \) for each value of \( x \)
Now, we will compute \( 3^x - 1 \) for each value of \( x \):

- For \( x = 1 \):
\[ 3^1 - 1 = 3 - 1 = 2 \quad (\text{even}) \]

- For \( x = 2 \):
\[ 3^2 - 1 = 9 - 1 = 8 \quad (\text{even}) \]

- For \( x = 3 \):
\[ 3^3 - 1 = 27 - 1 = 26 \quad (\text{even}) \]

- For \( x = 4 \):
\[ 3^4 - 1 = 81 - 1 = 80 \quad (\text{even}) \]

- For \( x = 5 \):
\[ 3^5 - 1 = 243 - 1 = 242 \quad (\text{even}) \]

### Step 3: Determine which values of \( x \) satisfy the condition
From our calculations, we see that for all values of \( x \) (1, 2, 3, 4, 5), \( 3^x - 1 \) is even. Therefore, all these values are included in our set.

### Step 4: Write the set in roster form
The set of all positive integers \( x \) less than 6 for which \( 3^x - 1 \) is even can be expressed in roster form as:
\[ \{1, 2, 3, 4, 5\} \]

### Final Answer
Thus, the final answer is:
\[ \{1, 2, 3, 4, 5\} \]