## Solving the Recurrence RelationGiven the recurrence relation:\[ a_{n+1} = 7a_n - 10a_{n-1}, \quad n \geq 2 \]with the initial conditions:\[ a_1 = 10, \quad a_2 = 29 \]We are to find the general solution for this recurrence relation.### Recurrence RelationThe recurrence relation expresses $ a_{n+1} $ (the next term of the sequence) in terms of the two previous terms $ a_n $ and $ a_{n-1} $:\[ a_{n+1} = 7a_n - 10a_{n-1} \]### Initial ConditionsThe sequence is initialized with the following values:\[ a_1 = 10 \]\[ a_2 = 29 \]Knowing these initial conditions, the next terms can be computed step by step using these values.In subsequent educational content, we would explore:1. **Characterizing the Recurrence Relation:** Discuss the homogeneous linear recurrence relations with constant coefficients.2. **Solving the Characteristic Equation:** Show how to derive the characteristic equation $ x^2 - 7x + 10 = 0 $, and solve for the roots $ x_1 $ and $ x_2 $.3. **Form of the General Solution:** Explain the general solution form $ a_n = C_1x_1^n + C_2x_2^n $ based on the roots of the characteristic equation.4. **Determining Constants:** Use the initial conditions to solve for the constants $ C_1 $ and $ C_2 $.5. **Explicit Formula:** Present the explicit formula for $ a_n $ based on the constants found.By working through these steps, students will learn to solve similar recurrence relations and understand the principles behind such sequences.

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Answered: Solve the recurrence relation an+1 7an…

Math

Advanced Math

Solve the recurrence relation an+1 7an – 10an-1, n 2 2, given a1 10, a2 = 29.

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

## Solving the Recurrence Relation

Given the recurrence relation:

\[ a_{n+1} = 7a_n - 10a_{n-1}, \quad n \geq 2 \]

with the initial conditions:

\[ a_1 = 10, \quad a_2 = 29 \]

We are to find the general solution for this recurrence relation.

### Recurrence Relation

The recurrence relation expresses $ a_{n+1} $ (the next term of the sequence) in terms of the two previous terms $ a_n $ and $ a_{n-1} $:

\[ a_{n+1} = 7a_n - 10a_{n-1} \]

### Initial Conditions

The sequence is initialized with the following values:

\[ a_1 = 10 \]
\[ a_2 = 29 \]

Knowing these initial conditions, the next terms can be computed step by step using these values.

In subsequent educational content, we would explore:

1. **Characterizing the Recurrence Relation:**
Discuss the homogeneous linear recurrence relations with constant coefficients.

2. **Solving the Characteristic Equation:**
Show how to derive the characteristic equation $ x^2 - 7x + 10 = 0 $, and solve for the roots $ x_1 $ and $ x_2 $.

3. **Form of the General Solution:**
Explain the general solution form $ a_n = C_1x_1^n + C_2x_2^n $ based on the roots of the characteristic equation.

4. **Determining Constants:**
Use the initial conditions to solve for the constants $ C_1 $ and $ C_2 $.

5. **Explicit Formula:**
Present the explicit formula for $ a_n $ based on the constants found.

By working through these steps, students will learn to solve similar recurrence relations and understand the principles behind such sequences.

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