Let {an }n>o be a sequence defined by a = n² + 2n − 3 for every n>0 a) Find the firsts 3 (elements) elements of the sequence. b) Show that the sequence satisfies the recurrence relation An = 2an-1 − An-2 +2 for every n ≥ 2. c) Find a closed formula for the first difference of the sequence {an}n>0² i.e., Δαπ = an+1-an for every n > 0.
Let {an }n>o be a sequence defined by a = n² + 2n − 3 for every n>0 a) Find the firsts 3 (elements) elements of the sequence. b) Show that the sequence satisfies the recurrence relation An = 2an-1 − An-2 +2 for every n ≥ 2. c) Find a closed formula for the first difference of the sequence {an}n>0² i.e., Δαπ = an+1-an for every n > 0.
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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![Let \(\{a_n\}_{n \geq 0}\) be a sequence defined by \(a_n = n^2 + 2n - 3\) for every \(n \geq 0\).
a) Find the first 3 (elements) of the sequence.
b) Show that the sequence satisfies the recurrence relation
\[ a_n = 2a_{n-1} - a_{n-2} + 2 \]
for every \(n \geq 2\).
c) Find a closed formula for the first difference of the sequence \(\{a_n\}_{n \geq 0}\), i.e., \(\Delta a_n = a_{n+1} - a_n\) for every \(n \geq 0\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F140c7626-2888-4935-8928-dcb55a4fc57b%2F4bcf2d02-b162-47e2-8012-80318be93e3c%2Fuelcxf_processed.png&w=3840&q=75)
Transcribed Image Text:Let \(\{a_n\}_{n \geq 0}\) be a sequence defined by \(a_n = n^2 + 2n - 3\) for every \(n \geq 0\).
a) Find the first 3 (elements) of the sequence.
b) Show that the sequence satisfies the recurrence relation
\[ a_n = 2a_{n-1} - a_{n-2} + 2 \]
for every \(n \geq 2\).
c) Find a closed formula for the first difference of the sequence \(\{a_n\}_{n \geq 0}\), i.e., \(\Delta a_n = a_{n+1} - a_n\) for every \(n \geq 0\).
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