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Kenneth Rosen Edition 7th 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 $ ... years if no money is ever withdrawn. How much is in the account after $n$ years if no money has been withdrawn?
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

8
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kennethrosen
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Kenneth Rosen Edition 7th 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 $2.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

6
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Kenneth Rosen Edition 7th 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?
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

7
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Kenneth Rosen Edition 7th 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?
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

10
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Kenneth Rosen Edition 7th 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 $8.1?$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

7
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 8.2 Question 4 (Page No. 524)
Solve these recurrence relations together with the initial conditions given. $a_{n} = a_{n1}+ 6a_{n2} \:\text{for}\: n \geq 2, a_{0} = 3, a_{1} = 6$ $a_{n} = 7a_{n1}− 10a_{n2} \:\text{for}\: n \geq 2, a_{0} = 2, a_{1} = 1$ ... $a_{n+2} = −4a_{n+1} + 5a_{n} \:\text{for}\: n \geq 0, a_{0} = 2, a_{1} = 8$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

9
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discretemathematics
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Kenneth Rosen Edition 7th Exercise 8.2 Question 3 (Page No. 524)
Solve these recurrence relations together with the initial conditions given. $a_{n} = 2a_{n−1}\:\text{for}\: n \geq 1, a_{0} = 3$ $a_{n} = a_{n−1} \:\text{for}\: n \geq 1, a_{0} = 2$ ... $a_{n} = a_{n−2} /4 \:\text{for}\: n \geq 2, a_{0} = 1, a_{1} = 0$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

12
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Kenneth Rosen Edition 7th 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. $a_{n} = 3a_{n2}$ $a_{n} = 3$ $a_{n} = a^{2}_{n−1}$ $an = a_{n−1} + 2a_{n−3}$ $an = a_{n−1}/n$ $an = a_{n−1} + a_{n−2} + n + 3$ $a_{n} = 4a_{n−2} + 5a_{n−4} + 9a_{n−7}$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

9
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kennethrosen
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Kenneth Rosen Edition 7th 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} $ $a_{n} = a_{n−2}$ $a_{n} = a_{n−1} + n$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

18
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 8.1 Question 57 (Page No. 512)
Dynamic programming can be used to develop an algorithm for solving the matrixchain 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 ... algorithm from part $(D)$ has $O(n^{3})$ worstcase complexity in terms of multiplications of integers.
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

10
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Kenneth Rosen Edition 7th 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 ... case complexity in terms of the number of additions and comparisons of your algorithm from part $(C)$ is linear.
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

8
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Kenneth Rosen Edition 7th 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. $ $10, 8, 7, 25, 20, 30, 5.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

11
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Kenneth Rosen Edition 7th 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. $ $10, 8, 7, 25, 20, 30, 5.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

10
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Kenneth Rosen Edition 7th 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 $1.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

7
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Kenneth Rosen Edition 7th 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.
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

6
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Kenneth Rosen Edition 7th 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 $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... $a_{n}, \triangledown a_{n},\: \text{and}\: \triangledown^{2}a_{n}.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

7
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Kenneth Rosen Edition 7th 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 $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... in terms of $a_{n}, \triangledown a_{n}, \triangledown ^{2}a_{n},\dots, \triangledown^{k}a_{n}.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

6
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Kenneth Rosen Edition 7th 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 $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... $a_{n−2} = a_{n} − 2\triangledown a_{n} + \triangledown^{2}a_{n}.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

5
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Kenneth Rosen Edition 7th 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 $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... $a_{n−1} = a_{n} − \triangledown a_{n}.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

6
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Kenneth Rosen Edition 7th 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 $(k + 1)^{\text{st}}$ ... $a_{n} = 4.$ $a_{n} = 2n.$ $a_{n} = n^{2}.$ $a_{n} = 2^{n}.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

7
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kennethrosen
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21
Kenneth Rosen Edition 7th Exercise 8.1 Question 46 (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 $(k + 1)^{\text{st}}$ ... $a_{n} = 4.$ $a_{n} = 2n.$ $a_{n} = n^{2}.$ $a_{n} = 2^{n}.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

7
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kennethrosen
discretemathematics
counting
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22
Kenneth Rosen Edition 7th Exercise 8.1 Question 45 (Page No. 512)
Question $3845$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the FrameStewart algorithm for moving the disks from peg $1$ to peg $4$ so that no ... how the disks are moved. Show that $R(n)\: \text{is}\: O(\sqrt{n}2^{\sqrt{2n}}).$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

9
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kennethrosen
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23
Kenneth Rosen Edition 7th Exercise 8.1 Question 44 (Page No. 512)
Question $3845$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the FrameStewart algorithm for moving the disks from peg $1$ to peg ... number of moves required to solve the Reve's puzzle for all integers $n$ with $1 \leq n \leq 25.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

8
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 8.1 Question 43 (Page No. 512)
Question $3845$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the FrameStewart algorithm for moving the disks from peg $1$ to peg $4$ so that no disk is ... $R(n) = \displaystyle{}\sum_{i = 1}^{k} i2^{i−1} − (t_{k} − n)2^{k−1}.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

9
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kennethrosen
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25
Kenneth Rosen Edition 7th Exercise 8.1 Question 42 (Page No. 512)
Question $3845$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the FrameStewart algorithm for moving the disks from peg $1$ to peg $4$ so that no disk ... that if $k$ is as chosen in question $41,$ then $R(n) − R(n − 1) = 2^{k−1}.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

6
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Kenneth Rosen Edition 7th Exercise 8.1 Question 41 (Page No. 512)
Question $3845$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the FrameStewart algorithm for moving the disks from peg $1$ to peg $4$ so that no disk is ever on ... $R(0) = 0\:\text{and}\: R(1) = 1.$
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

9
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Kenneth Rosen Edition 7th Exercise 8.1 Question 40 (Page No. 512)
Question $3845$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the FrameStewart algorithm for moving the disks from peg $1$ to peg $4$ ... chosen so that the fewest moves are required, for $5$ disks. $6$ disks. $7$ disks. $8$ disks.
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

7
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Kenneth Rosen Edition 7th Exercise 8.1 Question 39 (Page No. 512)
Question $3845$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the FrameStewart algorithm for moving the disks from peg $1$ ... are moved. Show that the Reve's puzzle with four disks can be solved using nine, and no fewer, moves
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

7
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 8.1 Question 38 (Page No. 512)
Question $3845$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the FrameStewart algorithm for moving the disks from peg $1$ ... are moved. Show that the Reve's puzzle with three disks can be solved using five, and no fewer, moves.
asked
May 3
in
Combinatory
by
Lakshman Patel RJIT

8
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 8.1 Question 37 (Page No. 512)
Question $3337$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ ... Determine $J (100), J (1000),\: \text{and}\: J (10,000)$ from your formula for $J (n).$
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May 3
in
Combinatory
by
Lakshman Patel RJIT

6
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