Problem 20 · 2021 Math Kangaroo
Stretch
Algebra & Patterns
arithmetic-sequencework-backward
There are eight boxes in the strip shown. The numbers in adjacent boxes have sum a or a + 1, as shown. The numbers in the first box and the eighth box are both 2021. What is the value of a?
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Answer: E — 4045
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Hint 1 of 2
Step along the boxes: each next entry is the arch-sum minus the current entry.
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Hint 2 of 2
The arch-sums alternate a and a+1; chain from box 1 = 2021 to box 8 = 2021 and solve for a.
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Approach: walk the chain of sums and set box 8 = 2021
- Starting at box 1 = 2021 and using the alternating sums a, a+1, a, a+1, ..., each box is the previous sum minus the previous box.
- Walking across, box 8 works out to a - 2024.
- Setting box 8 = 2021 gives a - 2024 = 2021.
- So a = 4045, choice (E).
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