Find a closed formula for the sequence with recursive : 2 = 1 and a2 definition an 2an An – 2 with a1 1 An
Find a closed formula for the sequence with recursive : 2 = 1 and a2 definition an 2an An – 2 with a1 1 An
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement:**
Find a closed formula for the sequence with the recursive definition:
\[
a_n = 2a_{n-1} - a_{n-2}
\]
with initial conditions:
\[
a_1 = 1 \quad \text{and} \quad a_2 = 2
\]
**Solution:**
To solve this problem, we need to find a closed formula (also known as an explicit formula) for the sequence. A closed formula directly calculates the \(n\)-th term without needing to know the previous terms, unlike the given recursive formula.
The recursive formula provided defines each term based on the previous two terms. Our goal is to identify a pattern or use characteristic equations typical for solving linear homogeneous recurrence relations. The initial conditions help determine specific coefficients if needed. Consider employing methods such as generating functions or characteristic equations to derive the formula.
**Instructions:**
Calculate the closed formula for \(a_n\) based on the provided recursive relation and initial conditions. Use algebraic techniques to solve and verify your formula by calculating the first few terms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd579cd13-c30e-49fa-a2cf-9ec3d96c3b0b%2Fd1a4df23-049d-489a-b4b1-bd19efc1abe7%2Fox10t9c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a closed formula for the sequence with the recursive definition:
\[
a_n = 2a_{n-1} - a_{n-2}
\]
with initial conditions:
\[
a_1 = 1 \quad \text{and} \quad a_2 = 2
\]
**Solution:**
To solve this problem, we need to find a closed formula (also known as an explicit formula) for the sequence. A closed formula directly calculates the \(n\)-th term without needing to know the previous terms, unlike the given recursive formula.
The recursive formula provided defines each term based on the previous two terms. Our goal is to identify a pattern or use characteristic equations typical for solving linear homogeneous recurrence relations. The initial conditions help determine specific coefficients if needed. Consider employing methods such as generating functions or characteristic equations to derive the formula.
**Instructions:**
Calculate the closed formula for \(a_n\) based on the provided recursive relation and initial conditions. Use algebraic techniques to solve and verify your formula by calculating the first few terms.
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