Problem 9 · 1985 AJHSME
Medium
Fractions, Decimals & Percents
telescoping-product
The product of the 9 factors (1 − 1⁄2)(1 − 1⁄3)(1 − 1⁄4) ⋯ (1 − 1⁄10) =
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Answer: A — 1⁄10.
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Hint 1 of 2
Don't multiply nine messy fractions. First simplify ONE factor: 1 − 1⁄2 = 1⁄2, 1 − 1⁄3 = 2⁄3, 1 − 1⁄4 = 3⁄4… do you see the staircase forming?
Still stuck? Show hint 2 →
Hint 2 of 2
Each factor is (n − 1)⁄n, so every numerator is the SAME number as the denominator just before it. That's telescoping — line them up and watch the inside cancel like a chain of dominoes, leaving only the very first numerator and the very last denominator.
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Approach: telescope
- Each factor 1 − 1⁄n equals (n − 1)⁄n, so the product is (1⁄2)(2⁄3)(3⁄4) ⋯ (9⁄10).
- Now cancel down the chain: the 2 on top of the second cancels the 2 on the bottom of the first, the 3 cancels the 3, and so on. Everything in the middle disappears, leaving the first top (1) over the last bottom (10).
- = 1⁄10.
- Why this transfers: when each piece of a product (or sum) hands its denominator to the next piece's numerator, only the two ends survive. Recognizing this 'telescope' turns a 9-step grind into reading off two numbers.
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