Problem 9 · 2006 AMC 8
Easy
Algebra & Patterns
telescoping-product
What is the product of 32 × 43 × 54 × … × 20062005?
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Answer: C — 1003.
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Hint 1 of 2
Don't multiply anything yet! Write out just the first few fractions and look for numbers that appear on both a top and a bottom — those cancel.
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Hint 2 of 2
This is a telescoping product: every numerator (3, 4, 5, …) reappears as the next denominator and cancels. Only the very first bottom and very last top survive.
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Approach: telescoping — let the chain cancel itself
- Line them up: (3/2)(4/3)(5/4)…(2006/2005). The 3 on top of the first fraction cancels the 3 on the bottom of the second; the 4 cancels the 4, and so on all the way down.
- Everything in the middle wipes out. What's left is just the first denominator (2) and the last numerator (2006): 2006 ÷ 2 = 1003.
- You'll see it again: any product where each term's top equals the next term's bottom telescopes — the answer is simply (last top) ÷ (first bottom). Spotting the cancellation pattern saves you from a 2004-step multiplication.
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