AMC 8 · 2016 · #8

Grade 5 arithmeticpattern

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

pattern-recognitionsequences-arithmeticmulti-digit-arithmetic identify-subproblemspattern-recognition ↑ Prerequisites: multi-digit-arithmeticsequences-arithmetic

📏 Short solution 💡 2 insights

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Problem

Compute the value of the alternating sum 100-98+96-94+92-90+…+8-6+4-2, where the terms are consecutive even numbers from 100 down to 2 with signs alternating +,-,+,-,… starting from a positive 100.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

100

AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

Computing 50 signed terms one at a time is slow and error-prone. Tool #7 (Identify Subproblems) lets us regroup the long sum into (100-98)+(96-94)+…+(4-2), turning one giant problem into many tiny copies of the same subproblem. Tool #5 (Look for a Pattern) then spots that every single pair evaluates to 2, so the original sum collapses to 2+2+…+2. Tool #9 (Solve an Easier Related Problem) is held in reserve as a sanity check — we can verify the method on the much shorter sum 4-2 or 8-6+4-2 before trusting it on the full 50-term version.

1STEP 1

Group the terms into consecutive pairs (100-98)+(96-94)+…, so each pair is a positive even number minus the next-smaller one.

100-98+96-94+…+4-2 = (100-98) + (96-94) + … + (4-2)

2STEP 2

Work out the first few pairs: 100-98=2, 96-94=2, …, so every pair equals 2.

100-98=2, 96-94=2, 92-90=2, …, 4-2=2

3STEP 3

The even numbers 2 to 100 make 50 terms, and pairing two at a time gives 25 pairs.

terms =

\frac{100}{2}

= 50, pairs =

\frac{50}{2}

= 25

4STEP 4

So the sum is 25 groups of 2, which is 25 × 2 = 50 — choice (C).

2+2+…+2₂5 copies = 25 × 2 = 50 → (C)

Answer

A rough estimate confirms the size. The sum starts with +100 but is immediately knocked down by -98, +96 by -94, and so on — every positive is canceled to within 2 by the next term. So the running total grows by only about 2 per pair, not by hundreds. With 25 pairs, we expect something near 25 × 2 = 50 — exactly choice (C). Choices like 80 or 100 are far too big given how aggressively the negatives cancel the positives.

💡Key takeaway

Group the 50 numbers into 25 pairs; each pair quietly equals 2, so the whole sum is just 25 × 2 = 50.

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