Problem 8 · 2022 AMC 8
Medium
Fractions, Decimals & Percents
fraction-to-decimal
What is the value of
13 · 24 · 35 · … · 1820 · 1921 · 2022 ?
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Answer: B — 1/231.
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Hint 1 of 2
Never multiply all this out. Each fraction is (a number)/(that number + 2), so the lists of numerators and denominators are nearly the same — line them up and watch what cancels.
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Hint 2 of 2
Numerators: 1, 2, 3, …, 20. Denominators: 3, 4, …, 22. Everything from 3 to 20 appears in both and cancels — only the two ends survive.
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Approach: telescoping — the lists overlap, so almost everything cancels
- Insight: don't compute — collect. Each factor is (k)/(k+2), so the numerators run 1, 2, 3, …, 20 and the denominators run 3, 4, 5, …, 22. Every number from 3 to 20 shows up in both lists, so it cancels.
- Only the unmatched ends remain — 1 and 2 on top, 21 and 22 on the bottom: 1 · 221 · 22 = 2462 = 1231.
- You'll see this again: a long product (or sum) where each term overlaps the next is a telescope — line up the lists and only the few leftover ends matter. The shift here is 2, so two terms survive on each end.
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