Use the quadratic formula to find the two roots. Here is the quadratic formula: -b + Vb^2- 4ac/2a Show your work here: e. Substitute two roots, r1 and r2 into the equation an= a1r1^n+ a2r2^n Show your work here: f. Now substitute to find two equations, a0 and a1 Remember to use the equation you found from step e. show your work here:
Use the quadratic formula to find the two roots. Here is the quadratic formula: -b + Vb^2- 4ac/2a Show your work here: e. Substitute two roots, r1 and r2 into the equation an= a1r1^n+ a2r2^n Show your work here: f. Now substitute to find two equations, a0 and a1 Remember to use the equation you found from step e. show your work here:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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Parts (a), (b), (c) are solved. d,e,f,g,h,i needs to be solved
I also added all the photos.
(a) Given that,
an= 6* an-1+ 7* an-2
Compare this equation with,
an= c1 an-1 + c2* an-2
We get that,
c1= 6
and c2= 7
(b) r2-c1r-c2= 0
Put the value of c1 and c2 in the above equation,
r2-6*r-7= 0
(c) a*r2*b*r+c= 0
a= 1
b= -6
and c= -7
d. Use the
-b + Vb^2- 4ac/2a
Show your work here:
e. Substitute two roots, r1 and r2 into the equation an= a1r1^n+ a2r2^n
Show your work here:
f. Now substitute to find two equations, a0 and a1 Remember to use the equation you found from step e.
show your work here:
a0= 3 =
a1= 6 =
g. Add the two equations together find a1 and a2
a1=
a2=
h. What is the solution to the recurrence relations?
an=
i. Find the 10" term of the sequence, using the solution to the recurrence relation you just found.
Show your work here:
a10=
![### Solving the Recurrence Relation
**Problem Statement:**
Solve the recurrence relation. Given:
- \( a_0 = 3 \)
- \( a_1 = 6 \)
- \( a_n = 6a_{n-1} + 7a_{n-2} \)
Show your work in the space provided.
**Steps:**
a. **Find \( c_1 \) and \( c_2 \):**
The recurrence relation can be expressed as:
\[ a_n = 6a_{n-1} + 7a_{n-2} \]
b. **Substitute \( c_1 \) and \( c_2 \) into the following equation:**
\[ r^2 - c_1 r - c_2 = 0 \]
- **Values to be determined:**
- \( c_1 = \)
- \( c_2 = \)
c. **Identify \( a \), \( b \), and \( c \) in the quadratic equation.**
- \( a = \)
- \( b = \)
- \( c = \)
d. **Use the quadratic formula to find the two roots.**
Here is the quadratic formula:
\[ \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- **Solution Steps:**
- Substitute the values of \( a \), \( b \), and \( c \) to find the roots.
- Work to be shown in the space provided.
---
**Note:** Use the provided space in your worksheet to perform calculations and derive the final result for \( r \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e0da6ca-d4b9-43ae-9bdc-db6e1c4ba965%2F57fb947c-c2e3-4620-b71d-c7cb3470bc37%2Fappcwtr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving the Recurrence Relation
**Problem Statement:**
Solve the recurrence relation. Given:
- \( a_0 = 3 \)
- \( a_1 = 6 \)
- \( a_n = 6a_{n-1} + 7a_{n-2} \)
Show your work in the space provided.
**Steps:**
a. **Find \( c_1 \) and \( c_2 \):**
The recurrence relation can be expressed as:
\[ a_n = 6a_{n-1} + 7a_{n-2} \]
b. **Substitute \( c_1 \) and \( c_2 \) into the following equation:**
\[ r^2 - c_1 r - c_2 = 0 \]
- **Values to be determined:**
- \( c_1 = \)
- \( c_2 = \)
c. **Identify \( a \), \( b \), and \( c \) in the quadratic equation.**
- \( a = \)
- \( b = \)
- \( c = \)
d. **Use the quadratic formula to find the two roots.**
Here is the quadratic formula:
\[ \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- **Solution Steps:**
- Substitute the values of \( a \), \( b \), and \( c \) to find the roots.
- Work to be shown in the space provided.
---
**Note:** Use the provided space in your worksheet to perform calculations and derive the final result for \( r \).
![**Text on Educational Website:**
---
**e.** Substitute two roots, \( r_1 \) and \( r_2 \), into the equation \( a_n = \alpha_1 r_1^n + \alpha_2 r_2^n \).
**Show your work here:**
[Blank box for student input]
---
**f.** Now substitute to find two equations, \( a_0 \) and \( a_1 \). Remember to use the equation you found from step e.
**Show your work here:**
\[ a_0 = 3 \]
\[ a_1 = 6 \]
[Blank box for student input]
---
**g.** Add the two equations together to find \( a_1 \) and \( a_2 \).
\[ a_1 = \]
\[ a_2 = \]
[Blank space for student input]
---
**h.** What is the solution to the recurrence relations?
\[ a_n = \]
[Blank space for student input]
---
**i.** Find the 10th term of the sequence, using the solution to the recurrence relation you just found.
**Show your work here:**
\[ a_{10} = \]
[Blank box for student input]
---
This set of instructions guides students through substituting roots into a recurrence relation, determining initial conditions, combining equations, and ultimately solving for specific terms in a sequence. Each step offers space for students to show their calculations and reasoning.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e0da6ca-d4b9-43ae-9bdc-db6e1c4ba965%2F57fb947c-c2e3-4620-b71d-c7cb3470bc37%2F0gkn2cb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Text on Educational Website:**
---
**e.** Substitute two roots, \( r_1 \) and \( r_2 \), into the equation \( a_n = \alpha_1 r_1^n + \alpha_2 r_2^n \).
**Show your work here:**
[Blank box for student input]
---
**f.** Now substitute to find two equations, \( a_0 \) and \( a_1 \). Remember to use the equation you found from step e.
**Show your work here:**
\[ a_0 = 3 \]
\[ a_1 = 6 \]
[Blank box for student input]
---
**g.** Add the two equations together to find \( a_1 \) and \( a_2 \).
\[ a_1 = \]
\[ a_2 = \]
[Blank space for student input]
---
**h.** What is the solution to the recurrence relations?
\[ a_n = \]
[Blank space for student input]
---
**i.** Find the 10th term of the sequence, using the solution to the recurrence relation you just found.
**Show your work here:**
\[ a_{10} = \]
[Blank box for student input]
---
This set of instructions guides students through substituting roots into a recurrence relation, determining initial conditions, combining equations, and ultimately solving for specific terms in a sequence. Each step offers space for students to show their calculations and reasoning.
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