A sequence of integers `a_1+a_2++a_n` satisfies `a_(n+2)=a_(n+1)-a_nforngeq1` . Suppose the sum of first 999 terms is 1003 and the sum of the first 1003 terms is -99. Find the sum of the first 2002 terms.

Explanation

To solve the problem, we need to analyze the sequence defined by the recurrence relation \( a_{n+2} = a_{n+1} - a_n \) and the given sums of the first 999 and 1003 terms.

### Step-by-Step Solution:

1. **Understanding the Recurrence Relation**:
The recurrence relation \( a_{n+2} = a_{n+1} - a_n \) suggests a pattern in the sequence. We can compute a few terms based on initial values \( a_1 \) and \( a_2 \):
- \( a_3 = a_2 - a_1 \)
- \( a_4 = a_3 - a_2 = (a_2 - a_1) - a_2 = -a_1 \)
...