Consider the following recurrence relation: -{+ C(n) = if n = 0 n+ 3. C(n-1) if n > 0. Prove by induction that C(n) = 3+1 -2n-3 for all n ≥ 0. 4 30+1 (Induction on n.) Let f(n) = 2n-1 4 Base Case: If n = 0, the recurrence relation says that C(0) = 0, and the formula says that f(0) = Inductive Hypothesis: Suppose as inductive hypothesis that C(k-1)=f(k-1) Inductive Step: Using the recurrence relation, C(K) = k + 3 = k + 3 X C(k-1), by the second part of the recurrence relation 3k-1+12(k-1) - 3 by inductive hypothesis X 0+1 for some k > 0. -2.0-3 , so they match.
Consider the following recurrence relation: -{+ C(n) = if n = 0 n+ 3. C(n-1) if n > 0. Prove by induction that C(n) = 3+1 -2n-3 for all n ≥ 0. 4 30+1 (Induction on n.) Let f(n) = 2n-1 4 Base Case: If n = 0, the recurrence relation says that C(0) = 0, and the formula says that f(0) = Inductive Hypothesis: Suppose as inductive hypothesis that C(k-1)=f(k-1) Inductive Step: Using the recurrence relation, C(K) = k + 3 = k + 3 X C(k-1), by the second part of the recurrence relation 3k-1+12(k-1) - 3 by inductive hypothesis X 0+1 for some k > 0. -2.0-3 , so they match.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 56EQ
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