Problem 29 · 2022 Math Kangaroo
Stretch
Algebra & Patterns
substitutionwork-backward
A sequence \(\langle a_n\rangle\) has \(0
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Answer: D — 4
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Hint 1 of 2
Use the recursions to write a3 and a7 in terms of a2 and a1.
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Hint 2 of 2
Also a2 itself satisfies a2 = a2·a1 + 1 — that extra equation pins a2 down.
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Approach: chain the recursion and use a2's own equation
- a3 = a2·a1 − 2 and a7 = a2·a3 − 2 = a2^2·a1 − 2a2 − 2 = 2.
- From a2 = a2·a1 + 1 we get a1 = 1 − 1/a2; substitute to get a2^2 − 3a2 − 4 = 0.
- So (a2−4)(a2+1) = 0; only a2 = 4 keeps 0 < a1 < 1.
- Thus a2 = 4.
Mark:
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