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FIT2004 Algorithms and data structures - S2 2026

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A sequence satisfies a(1) = 1a(1) = 1 and a(n) = 3 \cdot a(n-1)a(n) = 3 \cdot a(n-1) for n \ge 2n \ge 2. Which closed form matches it?
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Consider an algorithm that makes about \log n\log n levels of recursion, doing O(n)O(n) work in total at each level (this is merge sort). What is its overall time complexity?
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A sequence is defined by a(1) = 10a(1) = 10 and a(n) = a(n-1) + -1a(n) = a(n-1) + -1 for n \ge 2n \ge 2. What is a(7)a(7)?
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A sequence is defined by a(1) = 3a(1) = 3 and a(n) = 2 \cdot a(n-1)a(n) = 2 \cdot a(n-1) for n \ge 2n \ge 2. What is a(3)a(3)?
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Is the sequence defined by a(1) = 2,\ a(n) = 3 \cdot a(n-1)a(1) = 2,\ a(n) = 3 \cdot a(n-1) for n \ge 2n \ge 2 arithmetic, geometric, or neither?
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A divide-and-conquer algorithm has a 'split' step (dividing the input into subproblems) and a 'combine' step (assembling the result from the solved subproblems). Call a step 'trivial' if it only partitions or concatenates by position, without comparing or rearranging elements, and 'non-trivial' if it does the real work of comparing/rearranging elements. For quicksort, characterise the split and the combine step.
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An arithmetic series has first term 22, common difference 44, and 66 terms. What is the sum of all 66 terms?
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What is \log_2(1024)\log_2(1024) (i.e. the power to which 2 must be raised to give 10241024)?
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What is the worst-case time complexity of bubble sort of nn items?
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A loop runs ii from 1 to 88. For each ii, an inner loop runs jj from 1 to
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i, executing one statement. Exactly how many times does that statement execute in total?
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