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151
Kenneth Rosen Edition 7 Exercise 8.2 Question 20 (Page No. 525)
Find the general form of the solutions of the recurrence relation $a_{n} = 8a_{n−2} − 16a_{n−4}.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 19 (Page No. 525)
Solve the recurrence relation $a_{n} = −3a_{n−1} − 3a_{n−2} − a_{n−3}\:\text{with}\: a_{0} = 5, a_{1} = −9,\:\text{and}\: a_{2} = 15.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 18 (Page No. 525)
Solve the recurrence relation $a_{n} = 6a_{n−1} − 12a_{n−2} + 8a_{n−3} \:\text{with}\: a_{0} = −5, a_{1} = 4,\: \text{and}\: a_{2} = 88.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 17 (Page No. 525)
Prove this identity relating the Fibonacci numbers and the binomial coefficients: $f_{n+1} = C(n, 0) + C(n − 1, 1) +·\dots+ C(n − k, k),$ where $n$ is a positive integer and $k = n/2 .$ ... Show that the sequence $\{a_{n}\}$ satisfies the same recurrence relation and initial conditions satisfied by the sequence of Fibonacci numbers.]
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Kenneth Rosen Edition 7 Exercise 8.2 Question 16 (Page No. 525)
Prove Theorem $3:$ Let $c_{1},c_{2},\dots,c_{k}$ be real numbers. Suppose that the characteristic equation $r^{k}-c_{1}r^{k-1}-\dots - c_{k} = 0$ has $k$ distinct roots $r_{1},r_{2},\dots r_{k}.$ Then a sequence $\{a_{n}\}$ ... $n = 0,1,2,\dots,$ where $\alpha_{1},\alpha_{2},\dots,\alpha_{k}$ are constants.
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Kenneth Rosen Edition 7 Exercise 8.2 Question 15 (Page No. 525)
Find the solution to $a_{n} = 2a_{n−1} + 5a_{n−2} − 6a_{n−3}\: \text{with}\: a_{0} = 7, a_{1} = −4,\:\text{and}\: a_{2} = 8.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 14 (Page No. 525)
Find the solution to $a_{n} = 5a_{n−2}− 4a_{n−4} \:\text{with}\: a_{0} = 3, a_{1} = 2, a_{2} = 6, \:\text{and}\: a_{3} = 8.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 13 (Page No. 525)
Find the solution to $a_{n} = 7a_{n−2} + 6a_{n−3}\:\text{with}\: a_{0} = 9, a_{1} = 10, \text{and}\: a_{2} = 32.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 12 (Page No. 525)
Find the solution to $a_{n} = 2a_{n−1} + a_{n−2} − 2a_{n−3} \:\text{for}\: n = 3, 4, 5,\dots, \:\text{with}\: a_{0} = 3, a_{1} = 6, \:\text{and}\: a_{2} = 0.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 11 (Page No. 525)
The Lucas numbers satisfy the recurrence relation $L_{n} = L_{n−1} + L_{n−2},$ and the initial conditions $L_{0} = 2$ and $L_{1} = 1.$ Show that $L_{n} = f_{n−1} + f_{n+1}\: \text{for}\: n = 2, 3,\dots,$ where fn is the $n^{\text{th}}$ Fibonacci number. Find an explicit formula for the Lucas numbers.
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Kenneth Rosen Edition 7 Exercise 8.2 Question 9 (Page No. 525)
A deposit of $\$100,000$ is made to an investment fund at the beginning of a year. On the last day of each year two dividends are awarded. The first dividend is $ ... $n$ years if no money has been withdrawn?
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Kenneth Rosen Edition 7 Exercise 8.2 Question 8 (Page No. 524 - 525)
A model for the number of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years. Find a recurrence relation for $\{L_{n}\},$ ... if $100,000$ lobsters were caught in year $1\:\text{ and}\: 300,000$ were caught in year $2.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 7 (Page No. 524)
In how many ways can a $2 \times n$ rectangular checkerboard be tiled using $1 \times 2 \:\text{and}\: 2 \times 2$ pieces?
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Kenneth Rosen Edition 7 Exercise 8.2 Question 6 (Page No. 524)
How many different messages can be transmitted in $n$ microseconds using three different signals if one signal requires $1$ microsecond for transmittal, the other two signals require $2$ microseconds each for transmittal, and a signal in a message is followed immediately by the next signal?
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Kenneth Rosen Edition 7 Exercise 8.2 Question 5 (Page No. 524)
How many different messages can be transmitted in $n$ microseconds using the two signals described in question $19$ in Section $8.1?$
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May 3, 2020
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Kenneth Rosen Edition 7 Exercise 8.2 Question 4 (Page No. 524)
Solve these recurrence relations together with the initial conditions given. $a_{n} = a_{n-1}+ 6a_{n-2} \:\text{for}\: n \geq 2, a_{0} = 3, a_{1} = 6$ $a_{n} = 7a_{n-1}− 10a_{n-2} \:\text{for}\: n \geq 2, a_{0} = 2, a_{1} = 1$ ... $a_{n+2} = −4a_{n+1} + 5a_{n} \:\text{for}\: n \geq 0, a_{0} = 2, a_{1} = 8$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 3 (Page No. 524)
Solve these recurrence relations together with the initial conditions given. $a_{n} = 2a_{n−1}\:\text{for}\: n \geq 1, a_{0} = 3$ $a_{n} = a_{n−1} \:\text{for}\: n \geq 1, a_{0} = 2$ ... $a_{n} = a_{n−2} /4 \:\text{for}\: n \geq 2, a_{0} = 1, a_{1} = 0$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 2 (Page No. 524)
Determine which of these are linear homogeneous recurrence relations with constant coefficients. Also, find the degree of those that are. $a_{n} = 3a_{n-2}$ $a_{n} = 3$ $a_{n} = a^{2}_{n−1}$ $an = a_{n−1} + 2a_{n−3}$ $an = a_{n−1}/n$ $an = a_{n−1} + a_{n−2} + n + 3$ $a_{n} = 4a_{n−2} + 5a_{n−4} + 9a_{n−7}$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 1 (Page No. 524)
Determine which of these are linear homogeneous recurrence relations with constant coefficients. Also, find the degree of those that are. $a_{n} = 3a_{n−1} + 4a_{n−2} + 5a_{n−3}$ $a_{n} = 2na_{n−1} + a_{n−2}$ $a_{n} = a_{n−1} + a_{n−4}$ $a_{n} = a_{n−1} + 2 $ $a_{n} = a^{2}_{n−1} + a_{n−2} $ $a_{n} = a_{n−2}$ $a_{n} = a_{n−1} + n$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 57 (Page No. 512)
Dynamic programming can be used to develop an algorithm for solving the matrix-chain multiplication problem introduced in Section $3.3.$ This is the problem of determining how the product $A_{1}A_{2} \dots A_{n}$ can be computed ... algorithm from part $(D)$ has $O(n^{3})$ worst-case complexity in terms of multiplications of integers.
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Kenneth Rosen Edition 7 Exercise 8.1 Question 56 (Page No. 512)
In this question, we will develop a dynamic programming algorithm for finding the maximum sum of consecutive terms of a sequence of real numbers. That is, given a sequence of real numbers ... worst-case complexity in terms of the number of additions and comparisons of your algorithm from part $(C)$ is linear.
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Kenneth Rosen Edition 7 Exercise 8.1 Question 55 (Page No. 512)
For each part of question $54,$ use your algorithm from question $53$ to find the optimal schedule for talks so that the total number of attendees is maximized. $20, 10, 50, 30, 15, 25, 40.$ $100, 5, 10, 20, 25, 40, 30. $ $2, 3, 8, 5, 4, 7, 10. $ $10, 8, 7, 25, 20, 30, 5.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 54 (Page No. 512)
Use Algorithm $1$ to determine the maximum number of total attendees in the talks in Example $6$ if $w_{i},$ the number of attendees of talk $i, i = 1, 2,\dots, 7,$ is $20, 10, 50, 30, 15, 25, 40.$ $100, 5, 10, 20, 25, 40, 30. $ $2, 3, 8, 5, 4, 7, 10. $ $10, 8, 7, 25, 20, 30, 5.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 53 (Page No. 512)
Construct the algorithm described in the text after Algorithm $1$ for determining which talks should be scheduled to maximize the total number of attendees and not just the maximum total number of attendees determined by Algorithm $1.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 52 (Page No. 512)
Let $\{a_{n}\}$ be a sequence of real numbers. The backward differences of this sequence are defined recursively as shown next. The first difference $\triangledown a_{n}$ ... The resulting equation involving the sequences and its differences is called a difference equation.
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Kenneth Rosen Edition 7 Exercise 8.1 Question 51 (Page No. 512)
Let $\{a_{n}\}$ be a sequence of real numbers. The backward differences of this sequence are defined recursively as shown next. The first difference $\triangledown a_{n}$ is $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... $a_{n}, \triangledown a_{n},\: \text{and}\: \triangledown^{2}a_{n}.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 50 (Page No. 512)
Let $\{a_{n}\}$ be a sequence of real numbers. The backward differences of this sequence are defined recursively as shown next. The first difference $\triangledown a_{n}$ is $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... expressed in terms of $a_{n}, \triangledown a_{n}, \triangledown ^{2}a_{n},\dots, \triangledown^{k}a_{n}.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 49 (Page No. 512)
Let $\{a_{n}\}$ be a sequence of real numbers. The backward differences of this sequence are defined recursively as shown next. The first difference $\triangledown a_{n}$ is $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... $a_{n−2} = a_{n} − 2\triangledown a_{n} + \triangledown^{2}a_{n}.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 48 (Page No. 512)
Let $\{a_{n}\}$ be a sequence of real numbers. The backward differences of this sequence are defined recursively as shown next. The first difference $\triangledown a_{n}$ is $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... $a_{n−1} = a_{n} − \triangledown a_{n}.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 47 (Page No. 512)
Let $\{a_{n}\}$ be a sequence of real numbers. The backward differences of this sequence are defined recursively as shown next. The first difference $\triangledown a_{n}$ is $\triangledown a_{n} = a_{n} − a_{n−1}.$ The $(k + 1)^{\text{st}}$ ... $a_{n} = 4.$ $a_{n} = 2n.$ $a_{n} = n^{2}.$ $a_{n} = 2^{n}.$
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