Except for the first two numbers, every number in the sequence `-1, 3, -3,…..` is the product of the two immediately preceding numbers. How many numbers of this sequece are odd ?

Options

A. one
B. two
C. three
D. more than four

Answer

To solve the problem, we will analyze the given sequence step by step.

### Step 1: Understand the sequence
The sequence starts with the numbers -1 and 3. According to the problem, every subsequent number in the sequence is the product of the two immediately preceding numbers.

### Step 2: Write the first few terms of the sequence
1. The first two numbers are given:
- \( a_1 = -1 \)
- \( a_2 = 3 \)

2. The next number \( a_3 \) is calculated as:
\[ a_3 = a_1 \times a_2 = -1 \times 3 = -3 \]

3. The next number \( a_4 \) is calculated as:
\[ a_4 = a_2 \times a_3 = 3 \times -3 = -9 \]

4. The next number \( a_5 \) is calculated as:
\[ a_5 = a_3 \times a_4 = -3 \times -9 = 27 \]

5. The next number \( a_6 \) is calculated as:
\[ a_6 = a_4 \times a_5 = -9 \times 27 = -243 \]

### Step 3: List the sequence so far
The first six terms of the sequence are:
- \( a_1 = -1 \)
- \( a_2 = 3 \)
- \( a_3 = -3 \)
- \( a_4 = -9 \)
- \( a_5 = 27 \)
- \( a_6 = -243 \)

### Step 4: Identify the odd numbers
Now, we will check which of these numbers are odd:
- \( -1 \) is odd.
- \( 3 \) is odd.
- \( -3 \) is odd.
- \( -9 \) is odd.
- \( 27 \) is odd.
- \( -243 \) is odd.

### Step 5: Count the odd numbers
All the numbers in the sequence we have calculated are odd. Therefore, we can conclude that there are more than four odd numbers in this sequence.

### Final Answer
Thus, the answer to the question is that there are more than 4 odd numbers in the sequence.