Math Problem Statement
Find f(1), f(2), f(3), and, f(4) if f(n) is defined recursively by f(0) = 1 and for n = 0, 1, 2, . . . 1
ICS 141: Discrete Mathematics I (Fall 2014) a) f(n+1)=f(n)+2 f(1) = f(0 + 1) = f(0) + 2 = 1 + 2 = 3 f(2) = f(1 + 1) = f(1) + 2 = 3 + 2 = 5 f(3) = f(2 + 1) = f(2) + 2 = 5 + 2 = 7 f(4) = f(3 + 1) = f(3) + 2 = 7 + 2 = 9 b) f(n+1)=3f(n) f(1) = f(0 + 1) = 3f(0) = 3(1) = 3 f(2) = f(1 + 1) = 3f(1) = 3(3) = 9 f(3) = f(2 + 1) = 3f(2) = 3(9) = 27 f(4) = f(3 + 1) = 3f(3) = 3(27) = 81
Solution
Given the recursive function definitions for both cases (a) and (b), we can now compute , , , and .
Case (a) and :
We can compute the values step by step as follows:
Thus, for case (a), we have:
Case (b) and :
Similarly, compute the values step by step:
Thus, for case (b), we have:
Would you like any more details or explanations?
Related questions:
- What is the general solution for in case (a)?
- How would you solve a recursive function where ?
- Can you express case (b) in terms of a closed-form equation?
- How would the solution change if in either case?
- How do you approach finding recursive sequences in general?
Tip: When dealing with recursive functions, it's helpful to compute the first few terms to observe any patterns, which may help in formulating a closed form.
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Math Problem Analysis
Mathematical Concepts
Recursion
Arithmetic Progressions
Geometric Progressions
Formulas
f(n+1) = f(n) + 2 (Arithmetic Progression)
f(n+1) = 3f(n) (Geometric Progression)
Theorems
Recursion Formula
Suitable Grade Level
Grades 9-12