Problem 19 · 2016 AMC 8
Easy
Arithmetic & Operations
arithmetic-sequence
The sum of 25 consecutive even integers is 10,000. What is the largest of these 25 consecutive integers?
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Answer: E — 424.
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Hint 1 of 2
Evenly-spaced numbers are symmetric: every term below the center is balanced by an equal-sized term above it. So the AVERAGE of the whole list sits exactly on the MIDDLE term — you never have to write out all 25 numbers.
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Hint 2 of 2
Get the middle term as total ÷ count, then climb from the center to the largest: the top term is 12 steps of 2 above the middle.
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Approach: the middle term IS the average; then step out to the largest
- For 25 evenly-spaced numbers, the average equals the middle (13th) term: 10,000 ÷ 25 = 400.
- From the 13th term up to the 25th is 12 steps, each of size 2, so the largest = 400 + 12 × 2 = 424.
- Why this transfers: for any arithmetic sequence with an ODD number of terms, average = middle term — turning a sum problem into a one-line lookup. (With an even count, the average lands halfway between the two middle terms.)
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