AMC 8 · 2013 · #3

Grade 4 arithmeticpattern

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

pattern-recognitionsequences-arithmeticmental-arithmetic pattern-recognitionidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticorder-of-operations

📏 Short solution 💡 2 insights

📘 View easy version →

Problem

Find the value of 4 · (-1 + 2 - 3 + 4 - 5 + 6 - 7 + … + 1000), where the signs inside the parentheses alternate -, +, -, +, … from 1 all the way up to 1000.

Pick an answer.

(A)

-10

(B)

(C)

(D)

500

(E)

2000

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

Adding 1000 signed numbers one by one is hopeless — we need structure. Tool #5 (Look for a Pattern) is the perfect fit because the signs repeat in a clear two-step cycle (-, +, -, +, …), which means consecutive pairs of terms might collapse to something simple. Tool #7 (Identify Subproblems) makes that idea concrete: we split S into 500 small subproblems — each one is a pair (-odd) + (next even) — solve one of them, and reuse the result for all the others. Once S is known, the final 4 · S is a one-line multiplication.

1STEP 1

Group the 1000 terms into consecutive pairs — each shaped (negative odd) + (next even).

S = (-1 + 2) + (-3 + 4) + (-5 + 6) + … + (-999 + 1000)

2STEP 2

Compute a few pairs — every one collapses the same way: -(2k-1) + 2k = 1.

-1 + 2 = 1, -3 + 4 = 1, -5 + 6 = 1, …, -999 + 1000 = 1

3STEP 3

Count the pairs: 1000 terms ÷ 2 per pair = 500 pairs, so 500 ones get added.

number of pairs =

\frac{1000}{2}

= 500

4STEP 4

Sum the 500 ones to get S = 500, then multiply by the outside 4: 4 · 500 = 2000.

S = 500 × 1 = 500, 4 · S = 4 × 500 = 2000 → (E)

Answer

2000

A sanity check: the 1000 terms split into 500 pairs that each sum to +1, so S = 500. Multiplying by 4 gives 2000, which matches choice (E). The answer is positive (good — every pair was positive), and its size (2000) sits at the top of the answer list, which makes sense because we are scaling a 500-pair sum by 4.

💡Key takeaway

This AMC 8 problem only needs Grade 4 pattern-finding — pair up the terms, notice each pair adds to 1, then multiply!

More like this