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1
Kenneth Rosen Edition 7th Exercise 8.2 Question 28 (Page No. 525)
Find all solutions of the recurrence relation $a_{n} = 2a_{n1} + 2n^{2}.$ Find the solution of the recurrence relation in part $(A)$ with initial condition $a_{1} = 4.$
asked
May 5
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Combinatory
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Lakshman Patel RJIT

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Kenneth Rosen Edition 7th Exercise 8.2 Question 27 (Page No. 525)
What is the general form of the particular solution guaranteed to exist by Theorem 6 of the linear nonhomogeneous recurrence relation $a_{n} = 8a_{n2}  16a_{n4} + F(n)$ if $F(n) = n^{3}?$ $F(n) = (2)^{n}?$ $F(n) = n2^{n}? $ $F(n) = n^{2}4^{n}?$ $F(n) = (n^{2}  2)(2)^{n}?$ $F(n) = n^{4}2^{n}?$ $F(n) = 2?$
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May 5
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Combinatory
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Lakshman Patel RJIT

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Kenneth Rosen Edition 7th Exercise 8.2 Question 26 (Page No. 525)
What is the general form of the particular solution guaranteed to exist by Theorem $6$ of the linear nonhomogeneous recurrence relation $a_{n} = 6a_{n1}  12a_{n2} + 8a_{n3} + F (n)$ if $F (n) = n^{2}?$ $F (n) = 2^{n}?$ $F (n) = n2^{n}?$ $F (n) = (2)^{n}?$ $F (n) = n^{2}2^{n}?$ $F (n) = n^{3}(2)^{n}?$ $F (n) = 3?$
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May 5
in
Combinatory
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Lakshman Patel RJIT

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Kenneth Rosen Edition 7th Exercise 8.2 Question 25 (Page No. 525)
Determine values of the constants $A$ and $B$ such that $a_{n} = A{n} + B$ is a solution of recurrence relation $a_{n} = 2a_{n1} + n + 5.$ Use Theorem $5$ to find all solutions of this recurrence relation. Find the solution of this recurrence relation with $a_{0} = 4.$
asked
May 5
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Combinatory
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Lakshman Patel RJIT

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Kenneth Rosen Edition 7th Exercise 8.2 Question 24 (Page No. 525)
Consider the nonhomogeneous linear recurrence relation $a_{n} = 2a_{n1} + 2^{n}.$ Show that $a_{n} = n2^{n}$ is a solution of this recurrence relation. Use Theorem $5$ to find all solutions of this recurrence relation. Find the solution with $a_{0} = 2.$
asked
May 5
in
Combinatory
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Lakshman Patel RJIT

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Kenneth Rosen Edition 7th Exercise 8.2 Question 23 (Page No. 525)
Consider the nonhomogeneous linear recurrence relation $a_{n} = 3a_{n1} + 2^{n}.$ Show that $a_{n} = 2^{n+1}$ is a solution of this recurrence relation. Use Theorem $5$ to find all solutions of this recurrence relation. Find the solution with $a_{0} = 1.$
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May 5
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Combinatory
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Lakshman Patel RJIT

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Kenneth Rosen Edition 7th 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 $n$ ... Show that the sequence $\{a_{n}\}$ satisfies the same recurrence relation and initial conditions satisfied by the sequence of Fibonacci numbers.]
asked
May 4
in
Combinatory
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Lakshman Patel RJIT

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Kenneth Rosen Edition 7th 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 $r^{k}c_{1}r^{k1}\dots  c_{k} = 0$ 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.
asked
May 4
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Combinatory
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Lakshman Patel RJIT

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

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Kenneth Rosen Edition 7th Exercise 8.2 Question 10 (Page No. 525)
Prove Theorem $2:$ Let $c_{1}$ and $c_{2}$ be real numbers with $c_{2}\neq 0.$ Suppose that $r^{2}c_{1}rc_{2} = 0$ has only one root $r_{0}.$ A sequence $\{a_{n}\}$ ... $n = 0,1,2,\dots,$ where $\alpha_{1}$ and $\alpha_{2}$ are constants.
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May 3
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Combinatory
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Lakshman Patel RJIT

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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?
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May 3
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Combinatory
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Lakshman Patel RJIT

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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 $2.$
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May 3
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Combinatory
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Lakshman Patel RJIT

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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?
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May 3
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Lakshman Patel RJIT

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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?
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May 3
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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
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Lakshman Patel RJIT

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16
Kenneth Rosen Edition 7th Exercise 8.2 Question 4 (Page No. 524)
Solve these recurrence relations together with the initial conditions given. $a_{n} = a_{n1}+ 6a_{n2} \:\text{for}\: n \geq 2, a_{0} = 3, a_{1} = 6$ $a_{n} = 7a_{n1}− 10a_{n2} \:\text{for}\: n \geq 2, a_{0} = 2, a_{1} = 1$ ... $a_{n+2} = −4a_{n+1} + 5a_{n} \:\text{for}\: n \geq 0, a_{0} = 2, a_{1} = 8$
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May 3
in
Combinatory
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Lakshman Patel RJIT

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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
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Combinatory
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Lakshman Patel RJIT

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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.
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May 3
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Combinatory
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Lakshman Patel RJIT

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19
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. $ $10, 8, 7, 25, 20, 30, 5.$
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May 3
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Lakshman Patel RJIT

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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. $ $10, 8, 7, 25, 20, 30, 5.$
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May 3
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Combinatory
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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
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Combinatory
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Lakshman Patel RJIT

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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.
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May 3
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Combinatory
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Lakshman Patel RJIT

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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 $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... $a_{n}, \triangledown a_{n},\: \text{and}\: \triangledown^{2}a_{n}.$
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May 3
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Combinatory
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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
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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 $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... $a_{n−2} = a_{n} − 2\triangledown a_{n} + \triangledown^{2}a_{n}.$
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May 3
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Combinatory
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Lakshman Patel RJIT

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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 $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... $a_{n−1} = a_{n} − \triangledown a_{n}.$
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May 3
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Combinatory
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Lakshman Patel RJIT

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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
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Combinatory
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Lakshman Patel RJIT

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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
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in
Combinatory
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Lakshman Patel RJIT

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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 $R(n)\: \text{is}\: O(\sqrt{n}2^{\sqrt{2n}}).$
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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 $1 \leq n \leq 25.$
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May 3
in
Combinatory
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Lakshman Patel RJIT

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