Problem 9 · 2013 AMC 8
Easy
Algebra & Patterns
powers-of-two
The Incredible Hulk can double the distance it jumps with each succeeding jump. If its first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will it first be able to jump more than 1 kilometer (1,000 meters)?
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Answer: C — 11th jump.
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Hint 1 of 2
Doubling each time means you're climbing powers of 2: 1, 2, 4, 8, …. The only catch is the off-by-one — jump 1 is 20, not 21. Watch which jump number lines up with which power.
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Hint 2 of 2
"Repeated doubling" is always powers of 2. The danger is the exponent offset; anchor it by checking a small case (jump 3 = 4 = 22) before extrapolating.
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Approach: powers of 2, watching the off-by-one
- Jump 1 = 1 = 20, jump 2 = 2 = 21, … so jump n = 2n−1 meters. (The exponent is one behind the jump number — that's the spot people slip.)
- Memorized landmark: 210 = 1024 is the first power of 2 over 1000 (since 29 = 512 falls short).
- Set n − 1 = 10, so the first jump past 1000 m is the 11th.
- Worth keeping: 210 ≈ 1000 is a handy anchor — every 10 doublings multiplies by roughly a thousand.
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