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n 1 2 3 4 n-1 n R(n) (Simplified sum) 2 4 7 11 R(n-1) R(n) Pattern

n 1 2 3 4 n-1 n R(n) (Simplified sum) 2 4 7 11 R(n-1) R(n) Pattern

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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Generalize the pattern and represent the generalization as a polynomial and in terms of a triangular number
**Problem 1** | \( n \) | \( R(n) \) (Simplified sum) | Pattern | |--------|---------------------------|---------| | 1 | 2 | | | 2 | 4 | | | 3 | 7 | | | 4 | 11 | | | ... | | | | \( n-1 \) | \( R(n-1) \) | | | \( n \) | \( R(n) \) | | In this table, we are examining the values of \( R(n) \) (Simplified sum) as \( n \) increases. The column titles are as follows: - \( n \): Represents the position number in the sequence. - \( R(n) \) (Simplified sum): Denotes the simplified sum at position \( n \). - Pattern: A column to identify the pattern in the simplified sums, which is currently empty and intended for analysis or future input. Let's note the simplified sums provided: - When \( n = 1 \), \( R(n) = 2 \). - When \( n = 2 \), \( R(n) = 4 \). - When \( n = 3 \), \( R(n) = 7 \). - When \( n = 4 \), \( R(n) = 11 \). The subsequent rows, \( n-1 \) and \( n \), maintain the generalized forms \( R(n-1) \) and \( R(n) \) respectively. The empty “Pattern” column is for identifying an underlying pattern or formula in the sums as \( n \) increases. For instance, students or researchers might notice that each term seems to be increasing by a value that is incrementally larger as \( n \) increases (analyzing the differences between the sums can help identify the pattern). This table serves an educational purpose to help students understand sequences and series, and encourages them to spot patterns and derive formulas or rules.
Transcribed Image Text:**Problem 1** | \( n \) | \( R(n) \) (Simplified sum) | Pattern | |--------|---------------------------|---------| | 1 | 2 | | | 2 | 4 | | | 3 | 7 | | | 4 | 11 | | | ... | | | | \( n-1 \) | \( R(n-1) \) | | | \( n \) | \( R(n) \) | | In this table, we are examining the values of \( R(n) \) (Simplified sum) as \( n \) increases. The column titles are as follows: - \( n \): Represents the position number in the sequence. - \( R(n) \) (Simplified sum): Denotes the simplified sum at position \( n \). - Pattern: A column to identify the pattern in the simplified sums, which is currently empty and intended for analysis or future input. Let's note the simplified sums provided: - When \( n = 1 \), \( R(n) = 2 \). - When \( n = 2 \), \( R(n) = 4 \). - When \( n = 3 \), \( R(n) = 7 \). - When \( n = 4 \), \( R(n) = 11 \). The subsequent rows, \( n-1 \) and \( n \), maintain the generalized forms \( R(n-1) \) and \( R(n) \) respectively. The empty “Pattern” column is for identifying an underlying pattern or formula in the sums as \( n \) increases. For instance, students or researchers might notice that each term seems to be increasing by a value that is incrementally larger as \( n \) increases (analyzing the differences between the sums can help identify the pattern). This table serves an educational purpose to help students understand sequences and series, and encourages them to spot patterns and derive formulas or rules.
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