Problem 16 · 2021 Math Kangaroo
Hard
Algebra & Patterns
arithmetic-sequence
An infinite list of numbers has the property that, for each positive integer n, the average of the first n terms is n. How many terms are there less than 2021?
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Answer: C — 1010
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Hint 1 of 2
If the average of the first n terms is n, what is their sum?
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Hint 2 of 2
Get a single term by subtracting consecutive sums.
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Approach: turn the average condition into the n-th term
- Average of first n is n means the sum of the first n terms is n².
- The n-th term is n² − (n−1)² = 2n − 1 (the odd numbers 1, 3, 5, …).
- We need 2n − 1 < 2021, i.e. n ≤ 1010, so 1010 terms.
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