$2 - 2 + 2 - 2 + \dots 101$ terms = $x$

### Concept & Strategy For an alternating series where identical values are repeatedly added and subtracted, the sum depends entirely on whether the total number of terms is even or odd. Pairs of terms will cancel each other out to equal zero. ### Step-by-Step Solution **Given:** * The series is $2 - 2 + 2 - 2 + \dots$ * The total number of terms is $101$. **Calculation / Deduction:** Let's group the terms in the sequence into adjacent pairs: $(2 - 2) + (2 - 2) + (2 - 2) + \dots$ Each pair evaluates to $0$. Since each pair requires 2 terms, we divide the total number of terms by 2 to find how many complete pairs exist: $101 \div 2 = 50$ pairs with a remainder of $1$. This means there are exactly $50$ pairs of $(2 - 2)$ which all sum to $0$. The remainder of $1$ signifies that there is exactly one unpaired term left at the very end of the sequence. Because the series starts with a positive $2$, the 101st term will also be a positive $2$. Sum $= 0 + 0 + \dots + 0 + 2 = 2$. ### Exam Strategy & Shortcut For any alternating series in the format $a - a + a - a \dots$, instantly look at the term count $N$. If $N$ is even, the sum is $0$. If $N$ is odd, the sum is the first term $a$. Since $101$ is odd and the first term is $2$, the answer is instantly $2$. ### Common Pitfall Students often glance at the alternating $2$s, assume they all cancel out into infinity, and mistakenly select $0$. This happens when they fail to note that $101$ is an odd number, leaving one uncancelled term. ### Final Answer Therefore, the correct answer is **2**.