Problem 29 · 2022 Math Kangaroo
Stretch
Algebra & Patterns
arithmetic-sequencework-backward
Some points are marked on a straight line. Renate marks a new point between every pair of adjacent points, then repeats this three more times. Now there are 225 points on the line. How many points were there at the start?
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Answer: C — 15
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Hint 1 of 2
Inserting a point between every adjacent pair takes n points to a new count; find that rule.
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Hint 2 of 2
Each pass sends n points to 2n - 1; apply it four times and work backward from 225.
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Approach: iterate the doubling-minus-one rule
- Marking a point between each adjacent pair turns n points into n + (n-1) = 2n - 1.
- Doing this four times: n to 2n-1 to 4n-3 to 8n-7 to 16n-15.
- Set 16n - 15 = 225, so 16n = 240 and n = 15; the answer is C.
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