Problem 29 · 2013 Math Kangaroo
Stretch
Algebra & Patterns
arithmetic-seriessubstitution
Julian builds a sequence with \(a_{1} = 1\) and \(a_{m+n} = a_{m} + a_{n} + mn\) for all positive integers m and n. Find \(a_{100}\).
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Answer: E — 5050
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Hint 1 of 2
Try small cases: compute a_2, a_3 from the rule and spot the pattern.
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Hint 2 of 2
The values match the triangular numbers.
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Approach: recognise the closed form
- a_{m+n}=a_m+a_n+mn with a_1=1 fits a_n = n(n+1)/2 (it satisfies the relation).
- Then a_100 = 100·101/2.
- = 5050, so E.
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