In the sequence 1, 1, 0, 1, −1, … the first two terms a 1 and a 2 are each 1. The third term is the difference of the previous two and a 3 = a 1 − a 2 holds true. The fourth one is the sum of the previous two with a 4 = a 2 + a 3 , the fifth is the difference a 5 = a 3 − a 4 , a 6 = a 4 + a 5 , and so on, as well as the alternating difference and the sum. How big is the sum of the first 100 terms of this sequence?

Math Kangaroo Student 2012 Problem 29: Algebra Solution

Problem 29 · 2012 Math Kangaroo Stretch

Algebra & Patterns arithmetic-sequence

In the sequence 1, 1, 0, 1, −1, … the first two terms a₁ and a₂ are each 1. The third term is the difference of the previous two and a₃ = a₁ − a₂ holds true. The fourth one is the sum of the previous two with a₄ = a₂ + a₃, the fifth is the difference a₅ = a₃ − a₄, a₆ = a₄ + a₅, and so on, as well as the alternating difference and the sum. How big is the sum of the first 100 terms of this sequence?

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Answer: B — 3

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Hint 1 of 2

Just generate terms, alternately subtracting then adding the previous two, until the pattern repeats.

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Hint 2 of 2

Find the period and the sum over one full period, then handle the leftover terms.

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Approach: find the period, then sum 100 terms

Listing terms gives \(1,1,0,1,-1,0,-1,-1,0,-1,1,0\), after which \(a_{13}=1, a_{14}=1\) repeat the start, so the period is 12.
One full period sums to \(0\), so the first \(96 = 8\times12\) terms contribute \(0\).
The remaining four terms \(a_{97},\dots,a_{100}\) match \(a_1,\dots,a_4 = 1,1,0,1\), summing to \(3\), choice B.

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