Created on 2024-06-20Asked by Liam Wilson (Solvelet student)
Solve the recurrence relation: an=3an−1−2,witha0=1.
Solution
To solve the recurrence relation an=3an−1−2 with a0=1: 1. **Find the first few terms of the sequence:** a1=3a0−2=3⋅1−2=1,a2=3a1−2=3⋅1−2=1,a3=3a2−2=3⋅1−2=1. 2. **Observe the pattern:** It appears that an=1 for all n≥0. 3. **Prove by induction:** Base case: a0=1. Inductive step: Assume ak=1 for some k≥0. Then, ak+1=3ak−2=3⋅1−2=1. Thus, an=1 for all n≥0. Therefore, the solution to the recurrence relation is an=1. Solved on Solvelet with Basic AI Model
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DefinitionRecurrence relations are equations that characterize sequences; each term of the sequence depends on the previous ones, so their relationship is defined by a function. For example, the Fibonacci sequence is defined by the recurrence relation Fn=Fn−1+Fn−2 with initial conditions F0=0 and F1=1.
