Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Understanding Recursive Sequences
#### Problem Statement:
Consider the following recursively defined sequence:
\[ t_k = t_{k-1} + 3k + 1, \text{ for each integer } k \geq 1 \]
\[ t_0 = 0 \]
The steps below begin an iterative process to guess an explicit formula for the sequence.
\[
\begin{align*}
t_0 &= 0 \\
t_1 &= t_0 + 3 \cdot 1 + 1 = 3 + 1 \\
t_2 &= t_1 + 3 \cdot 2 + 1 = (3 + 1) + 3 \cdot 2 + 1 = 3 + 3 \cdot 2 + 2 \\
t_3 &= t_2 + 3 \cdot 3 + 1 = (3 + 3 \cdot 2 + 2) + 3 \cdot 3 + 1 = 3 + 3 \cdot 2 + 3 \cdot 3 + 3
\end{align*}
\]
Continue the iteration process in a free response. Then guess a formula for \( t_n \) as a summation written in expanded form, and use Theorem 5.2.1 to write it as a single fraction whose denominator is 2. (Submit a file with a maximum size of 1 MB.)
### Explanation:
By following the recursive definition, we can see the pattern forming in the sequence calculations:
1. For \( t_0 \), the value is 0 by definition.
2. For \( t_1 \), we use \( t_0 + 3 \cdot 1 + 1 \), simplifying to 4.
3. For \( t_2 \), we use \( t_1 + 3 \cdot 2 + 1 \), which simplifies to 3 + 2(3) + 2, etc.
### Further Steps:
1. **Continue Iterating**: Continue this process for subsequent values of \( t_k \) to confirm the emerging pattern.
2. **Guessing a Formula**: Based on the patterns observed, propose a general formula for \( t_n \).
3. **Verification**: Use summation notation and properties provided by mathematical theorems (like Theorem](https://content.bartleby.com/qna-images/question/8268c271-8991-49a2-aed9-a02bde5bd8ab/b5afce0b-c449-4b7a-94b2-4b9cb1eb5e2c/bnsuvpo_processed.png)
Transcribed Image Text:### Understanding Recursive Sequences
#### Problem Statement:
Consider the following recursively defined sequence:
\[ t_k = t_{k-1} + 3k + 1, \text{ for each integer } k \geq 1 \]
\[ t_0 = 0 \]
The steps below begin an iterative process to guess an explicit formula for the sequence.
\[
\begin{align*}
t_0 &= 0 \\
t_1 &= t_0 + 3 \cdot 1 + 1 = 3 + 1 \\
t_2 &= t_1 + 3 \cdot 2 + 1 = (3 + 1) + 3 \cdot 2 + 1 = 3 + 3 \cdot 2 + 2 \\
t_3 &= t_2 + 3 \cdot 3 + 1 = (3 + 3 \cdot 2 + 2) + 3 \cdot 3 + 1 = 3 + 3 \cdot 2 + 3 \cdot 3 + 3
\end{align*}
\]
Continue the iteration process in a free response. Then guess a formula for \( t_n \) as a summation written in expanded form, and use Theorem 5.2.1 to write it as a single fraction whose denominator is 2. (Submit a file with a maximum size of 1 MB.)
### Explanation:
By following the recursive definition, we can see the pattern forming in the sequence calculations:
1. For \( t_0 \), the value is 0 by definition.
2. For \( t_1 \), we use \( t_0 + 3 \cdot 1 + 1 \), simplifying to 4.
3. For \( t_2 \), we use \( t_1 + 3 \cdot 2 + 1 \), which simplifies to 3 + 2(3) + 2, etc.
### Further Steps:
1. **Continue Iterating**: Continue this process for subsequent values of \( t_k \) to confirm the emerging pattern.
2. **Guessing a Formula**: Based on the patterns observed, propose a general formula for \( t_n \).
3. **Verification**: Use summation notation and properties provided by mathematical theorems (like Theorem
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