Competition · AMC preparation · step 4 of 4
AMC 8 · 2016 · #8
Grade 5 arithmeticpatternPick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Group the terms in pairs
Group the terms into consecutive pairs (100-98)+(96-94)+…, so each pair is a positive even number minus the next-smaller one.
Inserting parentheses to group +a-b pairs is exactly the Grade 5 "use parentheses in expressions" move — same value, easier shape.
5.OA.A.1Identify SubproblemsEvaluate the first few pairs
Work out the first few pairs: 100-98=2, 96-94=2, …, so every pair equals 2.
Spotting that every pair equals the same constant (2) is the Grade 4 "analyze a generated pattern" skill in action.
4.OA.C.5Look For A PatternCount the pairs
The even numbers 2 to 100 make 50 terms, and pairing two at a time gives 25 pairs.
Even numbers from 2 to 100 correspond one-to-one with 1, 2, 3, …, 50 — a Grade 4 pattern-counting argument.
4.OA.C.5Look For A PatternMultiply 25 by 2
So the sum is 25 groups of 2, which is 25 × 2 = 50 — choice (C).
Turning a repeated sum into a multiplication is the Grade 3 "products as equal groups" definition.
The whole expression 100-98+96-94+…+4-2 adds up to twenty-five 2's, that is 25 × 2.
▸ Why?
Grouped into side-by-side pairs the line reads (100-98)+(96-94)+…+(4-2), where every pair equals 2 and there are 25 such pairs.
▸ Why?
In each pair the second number is the even number 2 smaller than the first, and since (a-2)+2=a, taking a-2 away from a must leave exactly 2.
▸ Why?
The pairs cover every even number from 2 to 100, and those 50 numbers split two-at-a-time into 25 equal pairs.
▸ Why?
Matching each even number 2k with the counting number k lines 2,4,…,100 up one-for-one with 1,2,…,50, so there are 50 even numbers.
▸ Why?
Sorting the 50 numbers into groups of 2 makes 25 groups, because 25 equal groups of 2 are exactly what fill up 50.
▸ Why?
Since each subtraction is the same as adding, the whole line is one running sum, and reordering or bracketing a sum never changes its total, so pairing neighbours keeps the same value.
▸ Why?
Adding the same number 2 a total of 25 times is exactly what 25 × 2 means — twenty-five equal groups of 2.
Group the 50 numbers into 25 pairs; each pair quietly equals 2, so the whole sum is just 25 × 2 = 50.
- Group the terms in pairs
- Evaluate the first few pairs
- Count the pairs
- Multiply 25 by 2
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