AMC 10 · 2020 · #8

Grade 4 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

pattern-recognitionsequences-arithmeticmental-arithmetic pattern-recognitionidentify-subproblems ↑ Prerequisites: sequences-arithmetic

📏 Medium solution 💡 3 insights

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Problem

Compute 1 + 2 + 3 - 4 + 5 + 6 + 7 - 8 + … + 197 + 198 + 199 - 200. Three terms are added, then one subtracted — and the cycle repeats through 200.

Pick an answer.

(A)

9,800

(B)

9,900

(C)

10,000

(D)

10,100

(E)

10,200

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The sign pattern repeats every 4 terms — Tool #5 says: group terms by the repeating unit and compute each block. Tool #7 then breaks the full expression into the sum of 50 block values, and finally the sum of an arithmetic sequence (2, 10, 18, …). Tool #9 (easier) lets us verify the per-block formula on just 1, 2, 3 blocks before sweeping all 50. This combination avoids any algebraic variable.

1STEP 1

Group all 200 terms into blocks of four, so $\frac{200}{4}$ = 50 blocks.

(1+2+3-4)_block 1 + (5+6+7-8)_block 2 + … + (197+198+199-200)_block 50

2STEP 2

Compute the first few blocks to spot the pattern; the totals start 2, 10, 18.

1+2+3-4 = 2, 5+6+7-8 = 10, 9+10+11-12 = 18

3STEP 3

The totals rise by 8 each step, so the last block (k = 50) is 197+198+199-200 = 394.

Block k value = 8k - 6, Block 50 = 8 · 50 - 6 = 394

4STEP 4

Pair the 50 block values from both ends: each pair sums to 2 + 394 = 396, with 25 pairs.

S =

\frac{50}{2}

· (2 + 394) = 25 · 396

5STEP 5

Split 396 = 400 - 4: 25 · 396 = 25 · 400 - 25 · 4 = 10,000 - 100 = 9,900.

25 · 396 = 25 · 400 - 25 · 4 = 10000 - 100 = 9900

6STEP 6

9,900 matches choice (B).

9,900 → (B)

Answer

9,900

Quick sanity check by another grouping. The full alternating expression equals (1 + 2 + … + 200) - 2 · (4 + 8 + 12 + … + 200) — because every multiple of 4 appears with a minus sign instead of a plus, costing twice its value. Compute: Σ₁²⁰⁰ =

\frac{200 · 201}{2}

= 20100. Multiples of 4 up to 200: there are 50 of them, summing to 4 · (1 + 2 + … + 50) = 4 · 1275 = 5100. So the answer is 20100 - 2 · 5100 = 20100 - 10200 = 9900. ✓

💡Key takeaway

This AMC 10 problem only needs Grade 4 pattern-spotting you already know — group every 4 terms: (1+2+3-4) = 2, (5+6+7-8) = 10, (9+10+11-12) = 18, jumping by 8 each time. Add up the 50 block totals (last one is 394) to get 9,900.

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