AMC 8 · 2018 · #5

Grade 4 arithmeticpattern

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

sequences-arithmeticpattern-recognitionparity pattern-recognitionidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticsequences-arithmetic

📏 Medium solution 💡 3 insights

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Problem

Compute the value of the expression 1+3+5+…+2017+2019 - (2+4+6+…+2016+2018) — that is, add all odd numbers from 1 to 2019, then subtract all even numbers from 2 to 2018.

Pick an answer.

(A)

-1010

(B)

-1009

(C)

1008

(D)

1009

(E)

1010

AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The expression is huge, but it has a very regular structure: alternating +odd, -even. Tool #5 (Look for a Pattern) is the natural fit — re-group the terms as (1-2)+(3-4)+(5-6)+… and notice that every pair has the same value. Tool #9 (Solve an Easier Related Problem) confirms the pattern with a tiny version (e.g., 1+3+5 - 2 - 4) before trusting it on the big one. Tool #7 (Identify Subproblems) helps us split the work cleanly into: (a) the leftover term 2019, (b) the paired part, and (c) counting how many pairs there are.

1STEP 1

Test it small first: 1+3+5-2-4 regroups as (1-2)+(3-4)+5 = 3 — each pair matches and the last odd number is left with no partner.

1+3+5-2-4 = (1-2)+(3-4)+5 = -1-1+5 = 3

2STEP 2

Regroup the whole expression the same way: (1-2)+(3-4)+…+(2017-2018) with 2019 left unpaired at the very end.

[(1-2)+(3-4)+(5-6)+…+(2017-2018)] + 2019

3STEP 3

Each pair is (odd) - (next even) = odd - (odd+1) = -1, so every pair falls short by 1.

1-2=-1, 3-4=-1, 5-6=-1, …, 2017-2018=-1

4STEP 4

Count the pairs: each uses one even number from {2,4,…,2018}, and dividing by 2 gives {1,…,1009}, so there are 1009 pairs.

2018 ÷ 2 = 1009 pairs

5STEP 5

Finish in one line: subtract the 1009 pair-shortfalls from the leftover 2019, i.e. 2019 - 1009, which gives the answer.

2019 - 1009 = 1010 → (E)

Answer

1010

The odd sum has one extra term (2019) compared to the even sum, so we expect the answer to be near 2019 minus a correction equal to the number of paired "shortfalls." Matching pairs (1,2),(3,4),…,(2017,2018) each cost 1, and there are 1009 of them, leaving 2019-1009=1010. That is positive (because the extra odd term dominates) and just slightly above 1009, which matches choice (E) and rules out the negative choices (A), (B).

💡Key takeaway

This AMC 8 problem only needs Grade 4 pattern-finding and multi-digit subtraction you already know!

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