Problem 12 · 2005 AMC 8
Easy
Algebra & Patterns
arithmetic-sequenceaverage
Big Al the ape ate 100 delicious yellow bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many delicious bananas did Big Al eat on May 5?
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Answer: D — 32.
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Hint 1 of 2
Five days, each 6 more than the last — the amounts step up evenly. In any such evenly-stepping list, the middle day equals the average. So don't set up a big equation; find the average first.
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Hint 2 of 2
The middle term of an odd-length arithmetic sequence is its mean. From the middle, count steps of 6 outward to any day you want.
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Approach: the middle day is the average
- Total is 100 over 5 days, so the average per day is 100 ÷ 5 = 20. Because the increase is steady, that average lands exactly on the middle day, May 3.
- May 5 is two days past the middle, each day +6: 20 + 2·6 = 32.
- Why this transfers: for any arithmetic sequence with an odd count of terms, sum ÷ count gives the center term instantly — far faster than solving for the first term. (Note 30, choice C, is the trap for stopping one step short.)
Another way — name the middle day x:
- Let May 3 = x. The five days are x−12, x−6, x, x+6, x+12; the ±12 and ±6 cancel, so the sum is just 5x.
- 5x = 100 ⇒ x = 20, and May 5 = x+12 = 32. Centering the variable makes the symmetric terms cancel cleanly.
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