10. Let (an) be the sequence defined recursively by, a1 = –1, a2 = 7, for n > 3, an = аn-1 + 2ап-2. Show that for all n, an = 2" +3(-1)".
10. Let (an) be the sequence defined recursively by, a1 = –1, a2 = 7, for n > 3, an = аn-1 + 2ап-2. Show that for all n, an = 2" +3(-1)".
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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Use Inductive method:
![10. Let \((a_n)\) be the sequence defined recursively by,
\[
a_1 = -1,
\]
\[
a_2 = 7,
\]
for \(n \geq 3\), \(a_n = a_{n-1} + 2a_{n-2}\).
Show that for all \(n\), \(a_n = 2^n + 3(-1)^n\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F63f643a6-d37e-4098-8dc7-7802480db8dc%2F3fa87e2b-5410-43e8-9073-136132d0f3fc%2F8zikb8j_processed.png&w=3840&q=75)
Transcribed Image Text:10. Let \((a_n)\) be the sequence defined recursively by,
\[
a_1 = -1,
\]
\[
a_2 = 7,
\]
for \(n \geq 3\), \(a_n = a_{n-1} + 2a_{n-2}\).
Show that for all \(n\), \(a_n = 2^n + 3(-1)^n\).
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