Consider the recurrence relation аn 3 4 : аm -1 — 4: ап - 2 a. Find the general solution to the recurrence relation (beware the repeated root!) An b. Find the solution when ao = 1 and a1 = 3 An c. Find the solution when ao = 1 and a1 = 6 An =
Consider the recurrence relation аn 3 4 : аm -1 — 4: ап - 2 a. Find the general solution to the recurrence relation (beware the repeated root!) An b. Find the solution when ao = 1 and a1 = 3 An c. Find the solution when ao = 1 and a1 = 6 An =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.4: Fractional Expressions
Problem 69E
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Question
![**Consider the recurrence relation**
\[ a_n = 4 \cdot a_{n-1} - 4 \cdot a_{n-2} \]
**a. Find the general solution to the recurrence relation (beware the repeated root!)**
\[ a_n = \]
**b. Find the solution when \( a_0 = 1 \) and \( a_1 = 3 \)**
\[ a_n = \]
**c. Find the solution when \( a_0 = 1 \) and \( a_1 = 6 \)**
\[ a_n = \]
**Explanation:**
The problem involves solving a recurrence relation, which is an equation that recursively defines a sequence. Given the recurrence relation:
\[ a_n = 4 \cdot a_{n-1} - 4 \cdot a_{n-2} \]
The task is to find the general solution and particular solutions for given initial conditions.
- **a.** The general solution involves analyzing the characteristic equation of the recurrence relation and solving for its roots. Pay attention to repeated roots as they affect the form of the solution.
- **b.** Substitute the initial conditions \( a_0 = 1 \) and \( a_1 = 3 \) into the general solution to find this specific sequence.
- **c.** Similar to part b, substitute the initial conditions \( a_0 = 1 \) and \( a_1 = 6 \) into the general solution to determine this sequence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd579cd13-c30e-49fa-a2cf-9ec3d96c3b0b%2F89e46871-85a3-474a-9133-ce0f8eebc7ef%2F9r35o8p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Consider the recurrence relation**
\[ a_n = 4 \cdot a_{n-1} - 4 \cdot a_{n-2} \]
**a. Find the general solution to the recurrence relation (beware the repeated root!)**
\[ a_n = \]
**b. Find the solution when \( a_0 = 1 \) and \( a_1 = 3 \)**
\[ a_n = \]
**c. Find the solution when \( a_0 = 1 \) and \( a_1 = 6 \)**
\[ a_n = \]
**Explanation:**
The problem involves solving a recurrence relation, which is an equation that recursively defines a sequence. Given the recurrence relation:
\[ a_n = 4 \cdot a_{n-1} - 4 \cdot a_{n-2} \]
The task is to find the general solution and particular solutions for given initial conditions.
- **a.** The general solution involves analyzing the characteristic equation of the recurrence relation and solving for its roots. Pay attention to repeated roots as they affect the form of the solution.
- **b.** Substitute the initial conditions \( a_0 = 1 \) and \( a_1 = 3 \) into the general solution to find this specific sequence.
- **c.** Similar to part b, substitute the initial conditions \( a_0 = 1 \) and \( a_1 = 6 \) into the general solution to determine this sequence.
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