1 - 2 + 3 - 4 + ... - 200. Find the average of these 200 terms.

Concept: When a series alternates in sign, split it into two arithmetic progressions (APs) — one built from the terms carrying one sign pattern, one from the terms carrying the other — find each AP's sum using S = n/2[2a + (n − 1)d], add the two sums, then divide by the total number of terms to get the average.

Working:

1. The series 1 − 2 + 3 − 4 + ... + 200 has 200 terms. Group it into the positive odd terms 1 + 3 + 5 + ... + 199 (an AP with first term a = 1, common difference d = 2) and the negative even terms −2 − 4 − 6 − ... − 200 (an AP with first term a = −2, common difference d = −2); each AP has 100 terms.

2. Sum of the positive AP: S1 = (100/2)[2(1) + (100 − 1)(2)] = 50 × 200 = 10,000.

3. Sum of the negative AP: S2 = (100/2)[2(−2) + (100 − 1)(−2)] = 50 × (−202) = −10,100.

4. Total sum of all 200 terms = S1 + S2 = 10,000 + (−10,100) = −100.

5. Average = total sum ÷ number of terms = −100 ÷ 200 = −1/2.

Cross-check: Pairing consecutive terms directly gives the same result: (1 − 2) + (3 − 4) + ... + (199 − 200) has 100 pairs, and each pair equals −1, so the total is 100 × (−1) = −100 — matching the AP-based sum above.

Result: Average = −100/200 = −1/2.