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7. 1)Detine the sequence: an n²-2n+1 (n= 1,2,3, ...) recursively.

7. 1)Detine the sequence: an n²-2n+1 (n= 1,2,3, ...) recursively.

Advanced Engineering Mathematics
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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Certainly! Here's the transcription tailored for an educational website: --- ### Recursive and Non-Recursive Sequences **Problem 7:** 1) Define the sequence recursively: \( a_n = n^2 - 2n + 1 \) for \( n = 1, 2, 3, \ldots \) 2) Define the sequence recursively: \( a_n = (n+1)! \) for \( n = 1, 2, 3, \ldots \) 3) Find the non-recursive formula for \( f(n) \): - Initial condition: \( f(0) = 6 \) - Recursive formula: \( f(n) = f(n-1) + 15 \) for \( n \geq 1 \) 4) Find the non-recursive formula for \( f(n) \): - Initial condition: \( f(0) = 3 \) - Recursive formula: \( f(n) = -2f(n-1)/7 \) for \( n \geq 1 \) --- ### Explanations: - **Recursive Definition:** This involves defining a sequence based on previous terms. It's useful for defining sequences where a pattern can be observed. - **Non-Recursive Formula:** Finding a direct formula that describes the sequence without dependency on previous terms simplifies calculations, especially for large \( n \). Understand these processes by recognizing patterns and relationships within sequences, a fundamental skill in mathematical analysis and computer science.
Transcribed Image Text:Certainly! Here's the transcription tailored for an educational website: --- ### Recursive and Non-Recursive Sequences **Problem 7:** 1) Define the sequence recursively: \( a_n = n^2 - 2n + 1 \) for \( n = 1, 2, 3, \ldots \) 2) Define the sequence recursively: \( a_n = (n+1)! \) for \( n = 1, 2, 3, \ldots \) 3) Find the non-recursive formula for \( f(n) \): - Initial condition: \( f(0) = 6 \) - Recursive formula: \( f(n) = f(n-1) + 15 \) for \( n \geq 1 \) 4) Find the non-recursive formula for \( f(n) \): - Initial condition: \( f(0) = 3 \) - Recursive formula: \( f(n) = -2f(n-1)/7 \) for \( n \geq 1 \) --- ### Explanations: - **Recursive Definition:** This involves defining a sequence based on previous terms. It's useful for defining sequences where a pattern can be observed. - **Non-Recursive Formula:** Finding a direct formula that describes the sequence without dependency on previous terms simplifies calculations, especially for large \( n \). Understand these processes by recognizing patterns and relationships within sequences, a fundamental skill in mathematical analysis and computer science.
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