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Kenneth Rosen Edition 7th Exercise 8.3 Question 16 (Page No. 535)
Solve the recurrence relation for the number of rounds in the tournament described in question $14.$
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

238
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kennethrosen
discretemathematics
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Kenneth Rosen Edition 7th Exercise 8.3 Question 15 (Page No. 535)
How many rounds are in the elimination tournament described in question $14$ when there are $32$ teams?
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

87
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 8.3 Question 14 (Page No. 535)
Suppose that there are $n = 2^{k}$ teams in an elimination tournament, where there are $\frac{n}{2}$ games in the first round, with the $\frac{n}{2} = 2^{k1}$ winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

98
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Kenneth Rosen Edition 7th Exercise 8.3 Question 13 (Page No. 535)
Give a bigO estimate for the function $f$ in question $12$ if $f$ is an increasing function.
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

97
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discretemathematics
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Kenneth Rosen Edition 7th Exercise 8.3 Question 12 (Page No. 535)
Find $f (n)$ when $n = 3k,$ where $f$ satisfies the recurrence relation $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

66
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kennethrosen
discretemathematics
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Kenneth Rosen Edition 7th Exercise 8.3 Question 11 (Page No. 535)
Give a bigO estimate for the function $f$ in question $10$ if $f$ is an increasing function.
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

25
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kennethrosen
discretemathematics
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Kenneth Rosen Edition 7th Exercise 8.3 Question 10 (Page No. 535)
Find $f (n)$ when $n = 2^{k},$ where $f$ satisfies the recurrence relation $f (n) = f (n/2) + 1 \:\text{with}\: f (1) = 1.$
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

19
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kennethrosen
discretemathematics
counting
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0
votes
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Kenneth Rosen Edition 7th Exercise 8.3 Question 9 (Page No. 535)
Suppose that $f (n) = f (n/5) + 3n^{2}$ when $n$ is a positive integer divisible by $5, \:\text{and}\: f (1) = 4.$ Find $f (5)$ $f (125)$ $f (3125)$
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

18
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kennethrosen
discretemathematics
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Kenneth Rosen Edition 7th Exercise 8.3 Question 8 (Page No. 535)
Suppose that $f (n) = 2f (n/2) + 3$ when $n$ is an even positive integer, and $f (1) = 5.$ Find $f (2)$ $f (8)$ $f (64)$ $(1024)$
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

19
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kennethrosen
discretemathematics
counting
recurrencerelations
descriptive
0
votes
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10
Kenneth Rosen Edition 7th Exercise 8.3 Question 7 (Page No. 535)
Suppose that $f (n) = f (n/3) + 1$ when $n$ is a positive integer divisible by $3,$ and $f (1) = 1.$ Find $f (3)$ $f (27)$ $f (729)$
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

26
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Kenneth Rosen Edition 7th Exercise 8.3 Question 6 (Page No. 535)
How many operations are needed to multiply two $32 \times 32$ matrices using the algorithm referred to in Example $5?$
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

18
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 8.3 Question 5 (Page No. 535)
Determine a value for the constant C in Example $4$ and use it to estimate the number of bit operations needed to multiply two $64$bit integers using the fast multiplication algorithm.
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

18
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discretemathematics
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0
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Kenneth Rosen Edition 7th Exercise 8.3 Question 4 (Page No. 535)
Express the fast multiplication algorithm in pseudocode.
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

29
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14
Kenneth Rosen Edition 7th Exercise 8.3 Question 3 (Page No. 535)
Multiply $(1110)_{2} \:\text{and}\: (1010)_{2}$ using the fast multiplication algorithm.
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

22
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Kenneth Rosen Edition 7th Exercise 8.3 Question 2 (Page No. 535)
How many comparisons are needed to locate the maximum and minimum elements in a sequence with $128$ elements using the algorithm in Example $2$?
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

16
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kennethrosen
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0
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Kenneth Rosen Edition 7th Exercise 8.3 Question 1 (Page No. 535)
How many comparisons are needed for a binary search in a set of $64$ elements?
asked
May 10
in
Combinatory
by
Lakshman Patel RJIT

28
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Kenneth Rosen Edition 7th Exercise 8.2 Question 52 (Page No. 527)
Prove Theorem $6:$Suppose that $\{a_{n}\}$ satisfies the liner nonhomogeneous recurrence relation $a_{n} = c_{1}a_{n1} + c_{2}a_{n2} + \dots + c_{k}a_{nk} + F(n),$ where $c_{1}.c_{2},\dots,c_{k}$ ... solution of the form $n^{m}(p_{t}n^{t} + p_{t1}n^{t1} + \dots + p_{1}n + p_{0})s^{n}.$
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

32
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Kenneth Rosen Edition 7th Exercise 8.2 Question 51 (Page No. 527)
Prove Theorem $4:$ 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 $t$ distinct roots $r_{1},r_{2},\dots,r_{t}$ ... $\alpha_{i,j}$ are constants for $1 \leq i \leq t\:\text{and}\: 0 \leq j \leq m_{i}  1.$
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

19
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 8.2 Question 53 (Page No. 527)
Solve the recurrence relation $T (n) = nT^{2}(n/2)$ with initial condition $T (1) = 6$ when $n = 2^{k}$ for some integer $k.$ [Hint: Let $n = 2^{k}$ and then make the substitution $a_{k} = \log T (2^{k})$ to obtain a linear nonhomogeneous recurrence relation.]
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

18
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kennethrosen
discretemathematics
counting
recurrencerelations
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+1
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Kenneth Rosen Edition 7th Exercise 8.2 Question 50 (Page No. 527)
It can be shown that Cn, the average number of comparisons made by the quick sort algorithm (described in preamble to question $50$ in exercise $5.4),$ when sorting $n$ ... $48$ to solve the recurrence relation in part $(A)$ to find an explicit formula for $C_{n}.$
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

16
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kennethrosen
discretemathematics
counting
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Kenneth Rosen Edition 7th Exercise 8.2 Question 49 (Page No. 527)
Use question $48$ to solve the recurrence relation $(n + 1)a_{n} = (n + 3)a_{n1} + n, \:\text{for}\: n \geq 1, \:\text{with}\: a_{0} = 1$
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

16
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kennethrosen
discretemathematics
counting
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descriptive
0
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22
Kenneth Rosen Edition 7th Exercise 8.2 Question 48 (Page No. 526)
Some linear recurrence relations that do not have constant coefficients can be systematically solved. This is the case for recurrence relations of the form $f (n)a_{n} = g(n)a_{n1} + h(n).$ Exercises $4850$ illustrate this. Show that the recurrence relation ...
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

12
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23
Kenneth Rosen Edition 7th Exercise 8.2 Question 47 (Page No. 526)
A new employee at an exciting new software company starts with a salary of $\$ ... year of employment. Solve this recurrence relation to find her salary for her $n^{\text{th}}$ year of employment.
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

13
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Kenneth Rosen Edition 7th Exercise 8.2 Question 46 (Page No. 526)
Suppose that there are two goats on an island initially.The number of goats on the island doubles every year by natural reproduction, and some goats are either added or removed each year. Construct a recurrence relation for the number of goats ... assuming that n goats are removed during the $n^{\text{th}}$ year for each $n \geq 3.$
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

19
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Kenneth Rosen Edition 7th Exercise 8.2 Question 45 (Page No. 526)
Suppose that each pair of a genetically engineered species of rabbits left on an island produces two new pairs of rabbits at the age of $1$ month and six new pairs of rabbits at the age of $2$ months and every month afterward ... $n$ months after one pair is left on the island.
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

15
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Kenneth Rosen Edition 7th Exercise 8.2 Question 44 (Page No. 526)
(Linear algebra required ) Let $A_{n}$ be the $n \times n$ matrix with $2s$ on its main diagonal, $1s$ in all positions next to a diagonal element, and $0s$ everywhere else. Find a recurrence relation for $d_{n},$ the determinant of $A_{n}.$ Solve this recurrence relation to find a formula for $d_{n}.$
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

11
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27
Kenneth Rosen Edition 7th Exercise 8.2 Question 43 (Page No. 526)
Express the solution of the linear nonhomogenous recurrence relation $a_{n} = a_{n1} + a_{n2} + 1\:\text{for}\: n \geq 2 \:\text{where}\: a_{0} = 0\:\text{and}\: a_{1} = 1$ in terms of the Fibonacci numbers. [Hint: Let $b_{n} = a_{n + 1}$ and apply question $42$ to the sequence $b_{n}.]$
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

13
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 8.2 Question 42 (Page No. 526)
Show that if $a_{n} = a_{n1} + a_{n2}, a_{0} = s\:\text{and}\: a_{1} = t,$ where $s$ and $t$ are constants, then $a_{n} = sf_{n1} + tf_{n}$ for all positive integers $n.$
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

13
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kennethrosen
discretemathematics
counting
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descriptive
0
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Kenneth Rosen Edition 7th Exercise 8.2 Question 41 (Page No. 526)
Use the formula found in Example $4$ for $f_{n},$ the $n^{\text{th}}$ Fibonacci number, to show that fn is the integer closest to $\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}$ Determine for which $n\: f_{n}$ is ... for which $n\: f_{n}$ is less than $\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}.$
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

15
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kennethrosen
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Kenneth Rosen Edition 7th Exercise 8.2 Question 40 (Page No. 526)
Solve the simultaneous recurrence relations $a_{n} = 3a_{n1} + 2b_{n1}$ $b_{n} = a_{n1} + 2b_{n1}$ with $a_{0} = 1 \: \text{and}\: b_{0} = 2.$
asked
May 6
in
Combinatory
by
Lakshman Patel RJIT

25
views
kennethrosen
discretemathematics
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