Problem 16 · 2020 Math Kangaroo
Stretch
Algebra & Patterns
periodicity
The sequence \(f_n\) is given by \(f_1 = 1\), \(f_2 = 2\) and \(f_n = f_{n-1} \cdot f_{n+1}\) for \(n \ge 2\). How many of the first 2020 terms of this sequence are even numbers?
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Answer: B — 674
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Hint 1 of 2
Rearrange the rule to f(n+1) = f(n)/f(n−1) and list a few terms.
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Hint 2 of 2
The sequence repeats with a short period — find it.
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Approach: detect the period and count evens within it
- From the rule we get f(n+1) = f(n)/f(n−1), giving 1, 2, 2, 1, 1/2, 1/2, then repeating with period 6.
- Each period of 6 has exactly two even terms (the two 2's).
- 2020 = 336×6 + 4; the 336 periods give 672 evens and the leading 1,2,2,1 add 2 more, totalling 674.
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