A sequence is show below:
1,4,-4….
The first term is 1. Each even numbered term is 3 more than previous term and each odd numbered term after the first is (-1) times the previous term. What is the sum of the first 32 terms of the above sequence?

Options

A. `-1`
B. 0
C. 1
D. 3

Answer

To solve the problem, we will first identify the pattern in the sequence based on the rules provided and then calculate the sum of the first 32 terms.

### Step-by-Step Solution:

1. **Identify the First Few Terms**:
- The first term \( a_1 = 1 \).
- The second term \( a_2 = 4 \) (given).
- The third term \( a_3 = -4 \) (given).
- Now, we need to find the next terms using the rules provided.

2. **Calculate the Next Terms**:
- **Fourth Term**: \( a_4 = a_2 + 3 = 4 + 3 = 7 \).
- **Fifth Term**: \( a_5 = -1 \times a_4 = -1 \times 7 = -7 \).
- **Sixth Term**: \( a_6 = a_5 + 3 = -7 + 3 = -4 \).
- **Seventh Term**: \( a_7 = -1 \times a_6 = -1 \times -4 = 4 \).
- **Eighth Term**: \( a_8 = a_7 + 3 = 4 + 3 = 7 \).

3. **List the First Eight Terms**:
- The first eight terms are: \( 1, 4, -4, 7, -7, -4, 4, 7 \).

4. **Identify the Pattern**:
- Notice that the sequence appears to repeat every four terms:
- \( a_1 = 1 \)
- \( a_2 = 4 \)
- \( a_3 = -4 \)
- \( a_4 = 7 \)
- \( a_5 = -7 \)
- \( a_6 = -4 \)
- \( a_7 = 4 \)
- \( a_8 = 7 \)
- We can see that the pattern repeats after every four terms.

5. **Sum of the First Four Terms**:
- Sum of the first four terms:
\[ S_4 = a_1 + a_2 + a_3 + a_4 = 1 + 4 - 4 + 7 = 8. \]

6. **Calculate the Sum of the First 32 Terms**:
- Since the sequence repeats every four terms, the sum of the first 32 terms can be calculated as:
\[ S_{32} = 8 \times S_4 = 8 \times 8 = 64. \]

7. **Final Result**:
- The sum of the first 32 terms of the sequence is \( 64 \).