Problem 2 · 2018 AMC 8
Easy
Fractions, Decimals & Percents
fraction-to-decimal
What is the value of the product
(1 + 11) · (1 + 12) · (1 + 13) · (1 + 14) · (1 + 15) · (1 + 16) ?
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Answer: D — 7.
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Hint 1 of 2
Before multiplying anything, turn each "1 + a fraction" into a single fraction. Notice each one becomes a fraction whose top is exactly one more than its bottom — that's a pattern begging to chain.
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Hint 2 of 2
The technique is telescoping: when you line up fractions where each numerator matches the next denominator, almost everything cancels and only the very first bottom and very last top survive.
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Approach: rewrite and telescope
- First combine each factor: 1 + 1/n = (n+1)/n. So the product becomes 21 · 32 · 43 · 54 · 65 · 76 — a neat staircase of consecutive numbers.
- Each top cancels the next bottom (the 2 on top kills the 2 below, the 3 kills the 3…), leaving only the first denominator (1) and the last numerator (7): the answer is 7.
- You'll see it again: whenever a long product or sum has terms that pass a piece to their neighbor, look for telescoping — you almost never compute the whole chain, just the two surviving ends.
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