T(2) = 2T(1)+1 = 2+1=3; 2²−1=3 ✔️

February 22, 2026

T(2) = 2T(1)+1 = 2+1=3; 2²−1=3 ✔️

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Frank is debugging a recursive function that computes Fibonacci numbers. The function runs with time complexity T(n) = 2T(n−1) + 1, with T(1) = 1. What is the value of T(10)?
This recurrence solves to T(n) = 2ⁿ − 1.
Check: T(1) = 2¹ − 1 = 1 ✔️
T(3) = 2×3+1=7; 2³−1=7 ✔️
So T(10) = 2¹⁰ − 1 = 1024 − 1 = <<1024-1=1023>>1023.
Peter is training a machine learning model. The error rate decreases exponentially: E(t) = 100 × (0.95)^t, where t is training epochs. After how many full epochs will the error rate drop below 50%?

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