The sequence with a₁ = 2 and recurrence relation an = an−1 + (2n + 5) generates these first few entries: 2, 11, 22, 35, 50, ... Find a closed formula for an. an =
The sequence with a₁ = 2 and recurrence relation an = an−1 + (2n + 5) generates these first few entries: 2, 11, 22, 35, 50, ... Find a closed formula for an. an =
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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Question
![**Title:** Finding a Closed Formula for a Recurrence Sequence
**Introduction:**
In this exercise, we explore a sequence defined by an initial term and a recurrence relation. Our task is to derive a closed formula for the sequence.
**Sequence Definition:**
- The sequence begins with \( a_1 = 2 \).
- The recurrence relation is given by:
\[
a_n = a_{n-1} + (2n + 5)
\]
**Generated Sequence:**
Using the initial term and recurrence relation, the first few terms of the sequence are:
2, 11, 22, 35, 50, ...
**Objective:**
Find a closed formula for \( a_n \).
**Solution:**
- A closed formula means expressing \( a_n \) directly in terms of \( n \), without dependence on previous terms.
**Task:**
Determine the closed formula for the sequence values.
\[
a_n = \boxed{}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb97f1139-ddfd-4008-99cf-70dde3f1fabc%2F52519927-0436-4717-9b85-42785585a8cd%2Fszradnk_processed.png&w=3840&q=75)
Transcribed Image Text:**Title:** Finding a Closed Formula for a Recurrence Sequence
**Introduction:**
In this exercise, we explore a sequence defined by an initial term and a recurrence relation. Our task is to derive a closed formula for the sequence.
**Sequence Definition:**
- The sequence begins with \( a_1 = 2 \).
- The recurrence relation is given by:
\[
a_n = a_{n-1} + (2n + 5)
\]
**Generated Sequence:**
Using the initial term and recurrence relation, the first few terms of the sequence are:
2, 11, 22, 35, 50, ...
**Objective:**
Find a closed formula for \( a_n \).
**Solution:**
- A closed formula means expressing \( a_n \) directly in terms of \( n \), without dependence on previous terms.
**Task:**
Determine the closed formula for the sequence values.
\[
a_n = \boxed{}
\]
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