Solve the recurrence relation. Given: • ao = 5, • a₁ = 6, • an = 8an-1-12an-2 Remember: is O O -b± √b² - 4ac 2a a an (6)" + 6(2)" b an = −(6) + 6(2)" = с an = −(6)” + 6(3)n d an = −(8)” + 6(12)”
Solve the recurrence relation. Given: • ao = 5, • a₁ = 6, • an = 8an-1-12an-2 Remember: is O O -b± √b² - 4ac 2a a an (6)" + 6(2)" b an = −(6) + 6(2)" = с an = −(6)” + 6(3)n d an = −(8)” + 6(12)”
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![## Solving Recurrence Relations
### Given:
- \( a_0 = 5 \)
- \( a_1 = 6 \)
- \( a_n = 8a_{n-1} - 12a_{n-2} \)
### Formula to Remember:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
### Multiple Choice Options:
a. \( a_n = (6)^n + 6(2)^n \)
b. \( a_n = -(6)^n + 6(2)^n \)
c. \( a_n = -(6)^n + 6(3)^n \)
d. \( a_n = -(8)^n + 6(12)^n \)
### Explanation:
- The problem presents a second-order linear homogeneous recurrence relation with constant coefficients.
- The general solution involves finding the roots of the characteristic equation derived from the recurrence relation.
- Substitute these roots into the general form to find a particular solution that satisfies the initial conditions.
By solving this recurrence relation step-by-step and comparing it to each of the given options, students can determine the correct form of \( a_n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F693a4d61-2c19-48be-8851-904ece734a28%2Fb2cfa737-fcc3-4d41-8c49-51e5748d0cde%2Fp8nszm7_processed.png&w=3840&q=75)
Transcribed Image Text:## Solving Recurrence Relations
### Given:
- \( a_0 = 5 \)
- \( a_1 = 6 \)
- \( a_n = 8a_{n-1} - 12a_{n-2} \)
### Formula to Remember:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
### Multiple Choice Options:
a. \( a_n = (6)^n + 6(2)^n \)
b. \( a_n = -(6)^n + 6(2)^n \)
c. \( a_n = -(6)^n + 6(3)^n \)
d. \( a_n = -(8)^n + 6(12)^n \)
### Explanation:
- The problem presents a second-order linear homogeneous recurrence relation with constant coefficients.
- The general solution involves finding the roots of the characteristic equation derived from the recurrence relation.
- Substitute these roots into the general form to find a particular solution that satisfies the initial conditions.
By solving this recurrence relation step-by-step and comparing it to each of the given options, students can determine the correct form of \( a_n \).
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