Problem 18 · 2016 Math Kangaroo
Medium
Algebra & Patterns
arithmetic-seriessum-constraint
The two kangaroos Jump and Hop start at the same time from the same line in the same direction. Each jumps exactly once per second. Jump always jumps 6 m. Hop jumps 1 m, then 2 m, then 3 m, and so on. After how many jumps does Hop catch up with Jump?
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Answer: B — 11
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Hint 1 of 2
After n jumps Jump has gone 6n m; Hop has gone 1 + 2 + ... + n m.
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Hint 2 of 2
Set n(n+1)/2 = 6n and solve.
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Approach: equate the distances after n jumps
- After n jumps Jump is at 6n m and Hop is at 1 + 2 + ... + n = n(n+1)/2 m.
- Hop catches up when n(n+1)/2 = 6n, i.e. n + 1 = 12, so n = 11.
- Hop catches Jump after 11 jumps.
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