Problem 29 · 2023 Math Kangaroo
Stretch
Algebra & Patterns
off-by-onesubstitution
Several points are marked on a straight line. Renate marks another point between each pair of adjacent points. She repeats this process three more times. Now there are 225 points marked on this straight line. How many points were marked to start with?
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Answer: A — 15
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Hint 1 of 2
Adding a point between every adjacent pair turns n points into 2nā1.
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Hint 2 of 2
Apply that step four times and set the result to 225.
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Approach: iterate the doubling-minus-one map
- One round: n points become n + (nā1) = 2nā1.
- Four rounds from x give 16x ā 15.
- Set 16x ā 15 = 225, so 16x = 240 and x = 15.
- She started with 15 points.
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