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Consider the following recurrence: f(1) = 1; f(2n) = 2f(n) - 1, for n ≥ 1; f(2n + 1) = 2f(n) + 1, for n ≥ 1. Then, which of the following statements is/are TRUE?

1. f(2ⁿ − 1) = 2ⁿ − 1
2. f(2ⁿ) = 1
3. f(5·2ⁿ) = 2^(n+1) + 1
4. f(2ⁿ + 1) = 2ⁿ + 1

Correct answer: f(2ⁿ − 1) = 2ⁿ − 1

Solution

The statement is true because when substituting n with 2ⁿ - 1 into the recurrence relation, it can be shown through induction that f(2ⁿ - 1) consistently evaluates to 2ⁿ - 1, confirming the pattern established by the recurrence.

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