Competition · AMC preparation · step 4 of 4
AMC 8 · 2016 · #19
Grade 6 arithmeticalgebraPick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Twenty-five consecutive even integers is too many to list, but the structure is highly regular. Tool #9 (Easier Problem) — try 3 or 5 consecutive evens first — exposes the pattern that the sum is always (count) × (middle term). Tool #5 (Pattern) lets us use that fact to jump straight to the middle term: 10,000 ÷ 25 = 400. Then Tool #7 (Subproblems) splits the rest into two clean pieces — find the middle term, then step out 12 places (by +2 each) to reach the largest. This dodges Tool #13 (Algebra) entirely.
Try a shorter run of evens
Test tiny cases: 4+6+8 = 3×6 and 2+4+6+8+10 = 5×6. Pattern: for an odd count of evenly-spaced terms, sum = count × middle term.
Working a small case before tackling 25 terms shows why pairs above and below the middle cancel — a Grade 3 "properties of operations" idea.
3.OA.B.5Solve An Easier Related ProblemFind the middle term
25 terms is an odd count, so sum = 25 × middle term. Divide: 10,000 ÷ 25 = 400 is the middle term.
For an odd-length evenly-spaced list, the mean and the middle value are the same number — a Grade 6 statistics fact.
The middle term of the 25 consecutive even integers is the total 10,000 shared equally among 25, which is 400.
▸ Why?
The 25 evenly spaced terms add up to exactly 25 of the middle term, so splitting the total into 25 equal shares hands back one middle term.
▸ Why?
The terms fall into 12 mirror pairs around the middle, each pair totals two middle terms for 24 middle terms, and the lone middle term makes 25.
▸ Why?
In each mirror pair one term sits as far above the middle as its partner sits below, so the surplus on the high side exactly fills the gap on the low side and the pair balances to two middle terms.
▸ Why?
Handing that equal surplus from the high term over to fill the low term's gap leaves their combined amount untouched — rearranging how a whole is split between two parts does not change the whole.
▸ Why?
The high term is some number of size-2 steps above the middle and its mirror partner is that same number of size-2 steps below, so both stand the same distance from the middle.
▸ Why?
Sharing the total into 25 equal parts undoes multiplying one part by 25, because division reverses multiplication.
Locate the middle position
With 25 terms the middle is the 13th term — 12 sit on each side — so term 13 equals 400.
Splitting the count into "left of middle", "middle", "right of middle" is a Grade 4 sequence-position move.
4.OA.C.5Identify SubproblemsStep up to the largest term
Term 13 to term 25 is 12 steps of +2, so the largest = 400 + 24 = 424 (E).
Adding the common difference 12 times to walk from term 13 to term 25 is the same skip-counting Grade 4 students do with patterns.
4.OA.C.5Look For A PatternIf you have an odd number of evenly-spaced numbers, the average IS the middle one — a Grade 6 idea that cracks this AMC 8 problem in one division!
- Try a shorter run of evens
- Find the middle term
- Locate the middle position
- Step up to the largest term
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