Problem 27 · 2013 Math Kangaroo
Stretch
Algebra & Patterns
Number Theory
spiral-patternarithmetic-series
A sequence of numbers begins 1, −1, −1, 1, −1. Each new number is the product of the two numbers before it (for example, the sixth number is the product of the fourth and fifth). What is the sum of the first 2013 numbers?
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Answer: B — −671
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Hint 1 of 2
Compute a few more terms; the sequence soon repeats.
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Hint 2 of 2
Find the repeating block and its sum, then count how many blocks fit in 2013 terms.
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Approach: find the period and sum blocks
- The terms run 1, −1, −1, 1, −1, −1, 1, −1, −1, … — the block (1, −1, −1) repeats.
- Each block of 3 sums to −1.
- 2013 = 3 × 671, so there are 671 complete blocks.
- Total = 671 × (−1) = −671.
Mark:
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