Compute the values: D2-2D1, D3-3D2, D4-4D3, D5-5D4, …, D10-10D9, where the D’s are the derangements values described on P.335. What is the pattern?
Compute the values: D2-2D1, D3-3D2, D4-4D3, D5-5D4, …, D10-10D9, where the D’s are the derangements values described on P.335. What is the pattern?
Computer Networking: A Top-Down Approach (7th Edition)
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Compute the values: D2-2D1, D3-3D2, D4-4D3, D5-5D4, …, D10-10D9, where the D’s are the derangements values described on P.335. What is the pattern?
![**Examples Defined by Recurrence Equations**
---
Using Eq. 8.1.2 together with the values of \( D_1 \) and \( D_2 \), we can evaluate \( D_n \) for any value of \( n \):
\[
\begin{align*}
D_3 &= (3 - 1)\{D_2 + D_1\} = 2\{1 + 0\} = 2 \\
D_4 &= (4 - 1)\{D_3 + D_2\} = 3\{2 + 1\} = 9 \\
D_5 &= (5 - 1)\{D_4 + D_3\} = 4\{9 + 2\} = 44 \\
D_6 &= (6 - 1)\{D_5 + D_4\} = 5\{44 + 9\} = 265 \\
D_7 &= (7 - 1)\{D_6 + D_5\} = 6\{265 + 44\} = 1\,854 \\
D_8 &= (8 - 1)\{D_7 + D_6\} = 7\{1\,854 + 265\} = 14\,833 \\
D_9 &= (9 - 1)\{D_8 + D_7\} = 8\{14\,833 + 1\,854\} = 133\,496 \\
D_{10} &= (10 - 1)\{D_9 + D_8\} = 9\{133\,496 + 14\,833\} = 1\,334\,961 \\
\end{align*}
\]
---
// It’s strange that 1,334,961 equals \( 10 \times (133,496) + 1 \). Or is it?
// Is there a (convenient and compact) formula for \( D_n \) that we can use to calculate its values?
The sequence on \( P \) defined by \( S_n = A \times n! \) where \( A \) is any real number satisfies the recurrence equation (8.1.2). If \( n \geq 3 \) then:
\[
\begin{align*}
(n -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4792360-22b4-40c0-a48a-02ce03c9f6f7%2Fb386ee9b-9957-408e-a054-5b32ed3148d5%2Fx4z4u9_processed.png&w=3840&q=75)
Transcribed Image Text:**Examples Defined by Recurrence Equations**
---
Using Eq. 8.1.2 together with the values of \( D_1 \) and \( D_2 \), we can evaluate \( D_n \) for any value of \( n \):
\[
\begin{align*}
D_3 &= (3 - 1)\{D_2 + D_1\} = 2\{1 + 0\} = 2 \\
D_4 &= (4 - 1)\{D_3 + D_2\} = 3\{2 + 1\} = 9 \\
D_5 &= (5 - 1)\{D_4 + D_3\} = 4\{9 + 2\} = 44 \\
D_6 &= (6 - 1)\{D_5 + D_4\} = 5\{44 + 9\} = 265 \\
D_7 &= (7 - 1)\{D_6 + D_5\} = 6\{265 + 44\} = 1\,854 \\
D_8 &= (8 - 1)\{D_7 + D_6\} = 7\{1\,854 + 265\} = 14\,833 \\
D_9 &= (9 - 1)\{D_8 + D_7\} = 8\{14\,833 + 1\,854\} = 133\,496 \\
D_{10} &= (10 - 1)\{D_9 + D_8\} = 9\{133\,496 + 14\,833\} = 1\,334\,961 \\
\end{align*}
\]
---
// It’s strange that 1,334,961 equals \( 10 \times (133,496) + 1 \). Or is it?
// Is there a (convenient and compact) formula for \( D_n \) that we can use to calculate its values?
The sequence on \( P \) defined by \( S_n = A \times n! \) where \( A \) is any real number satisfies the recurrence equation (8.1.2). If \( n \geq 3 \) then:
\[
\begin{align*}
(n -
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