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
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## 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.
Transcribed Image Text:## 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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