d. By experiment, guess or derivation, find real numbers r and s so that An+2 + ran+1+ san = 0, bn+2+rbn+1+ sb, = 0 %3D and prove the relation. What is the connection between 3+ i and the polynomial X² + rX + s? Hint: there is nothing particularly special about 3+i.
d. By experiment, guess or derivation, find real numbers r and s so that An+2 + ran+1+ san = 0, bn+2+rbn+1+ sb, = 0 %3D and prove the relation. What is the connection between 3+ i and the polynomial X² + rX + s? Hint: there is nothing particularly special about 3+i.
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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The question is attached that relates to the original problem of
Let (3+i)^n=a_n +ib_n.
I found the r and s values to be -6 and 10. I just don't know how to answer the connection between 3+i and the polynomial X^2+rX+s. There needs to be a more general type of answer other than 3+i is one of the roots of the polynomial.
![**d.** By experiment, guess or derivation, find real numbers \( r \) and \( s \) so that
\[
a_{n+2} + r a_{n+1} + s a_n = 0, \quad b_{n+2} + r b_{n+1} + s b_n = 0
\]
and prove the relation. What is the connection between \( 3 + i \) and the polynomial \( X^2 + rX + s \)? *Hint: there is nothing particularly special about \( 3 + i \).*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e1e2e4-888b-4182-8c02-fd46dda7f6b1%2F094cd6f2-75e9-41c2-ab84-2b06f28fc6c6%2Fi1ze18o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**d.** By experiment, guess or derivation, find real numbers \( r \) and \( s \) so that
\[
a_{n+2} + r a_{n+1} + s a_n = 0, \quad b_{n+2} + r b_{n+1} + s b_n = 0
\]
and prove the relation. What is the connection between \( 3 + i \) and the polynomial \( X^2 + rX + s \)? *Hint: there is nothing particularly special about \( 3 + i \).*
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