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Powered by Question2AnswerKenneth Rosen Edition 7th Exercise 8.3 Question 11 (Page No. 535)
https://gateoverflow.in/339402/kenneth-rosen-edition-7th-exercise-8-question-11-page-no-535
Give a big-O estimate for the function $f$ in question $10$ if $f$ is an increasing function.Combinatoryhttps://gateoverflow.in/339402/kenneth-rosen-edition-7th-exercise-8-question-11-page-no-535Sat, 09 May 2020 21:33:13 +0000Kenneth Rosen Edition 7th Exercise 8.3 Question 6 (Page No. 535)
https://gateoverflow.in/339397/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 $5?$Combinatoryhttps://gateoverflow.in/339397/kenneth-rosen-edition-7th-exercise-8-3-question-6-page-no-535Sat, 09 May 2020 21:19:23 +0000Kenneth Rosen Edition 7th Exercise 8.3 Question 5 (Page No. 535)
https://gateoverflow.in/339396/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.Combinatoryhttps://gateoverflow.in/339396/kenneth-rosen-edition-7th-exercise-8-3-question-5-page-no-535Sat, 09 May 2020 21:17:29 +0000Kenneth Rosen Edition 7th Exercise 8.3 Question 3 (Page No. 535)
https://gateoverflow.in/339394/kenneth-rosen-edition-7th-exercise-8-3-question-3-page-no-535
Multiply $(1110)_{2} \:\text{and}\: (1010)_{2}$ using the fast multiplication algorithm.Combinatoryhttps://gateoverflow.in/339394/kenneth-rosen-edition-7th-exercise-8-3-question-3-page-no-535Sat, 09 May 2020 21:14:49 +0000Kenneth Rosen Edition 7th Exercise 8.3 Question 2 (Page No. 535)
https://gateoverflow.in/339393/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 $2$?Combinatoryhttps://gateoverflow.in/339393/kenneth-rosen-edition-7th-exercise-8-3-question-2-page-no-535Sat, 09 May 2020 21:10:58 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 52 (Page No. 527)
https://gateoverflow.in/339232/kenneth-rosen-edition-7th-exercise-8-question-52-page-no-527
Prove Theorem $6:$Suppose that $\{a_{n}\}$ satisfies the liner nonhomogeneous recurrence relation $$a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k} + F(n),$$ where $c_{1}.c_{2},\dots,c_{k}$ are real numbers , and<br />
<br />
$$F(n) = (b_{t}n^{t} + b_{t-1}n^{t-1}) + \dots + b_{1}n + b_{0})s^{n},$$ where $b_{0},b_{1},\dots,b_{t}$ and $s$ are real numbers. When $s$ is is not a root of the characteristic equation of the associated linear homogeneous recurrence relation, there is a particular solution of the form $$(p_{t}n^{t} + p_{t-1}n^{t-1} + \dots + p_{1}n + p_{0})s^{n}.$$<br />
<br />
When $s$ is a root of this characteristic equation and its multiplicity is $m,$ there is a particular solution of the form $$n^{m}(p_{t}n^{t} + p_{t-1}n^{t-1} + \dots + p_{1}n + p_{0})s^{n}.$$Combinatoryhttps://gateoverflow.in/339232/kenneth-rosen-edition-7th-exercise-8-question-52-page-no-527Wed, 06 May 2020 04:33:22 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 51 (Page No. 527)
https://gateoverflow.in/339231/kenneth-rosen-edition-7th-exercise-8-question-51-page-no-527
Prove Theorem $4:$ Let $c_{1},c_{2},\dots,c_{k}$ be real numbers. Suppose that the characteristic equation<br />
<br />
$$r^{k}-c_{1}r^{k-1}-\dots c_{k} = 0$$<br />
<br />
has $t$ distinct roots $r_{1},r_{2},\dots,r_{t}$ with multiplicities $m_{1},m_{2},\dots,m_{t},$ respectively, so that $m_{i}\geq 1\:\text{for}\: i = 1,2,\dots,t$ and $m_{1} + m_{2} + \dots + m_{t} = k.$ Then a sequence $\{a_{n}\}$ is a solution of the recurrence relation.<br />
<br />
$$a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k}$$<br />
<br />
if and only if<br />
<br />
$$a_{n} = (\alpha_{1},0 + \alpha_{1,1n} + \dots + \alpha_{1,m_{1}-1}n^{m_{1}-1})r_{1}^{n} + (\alpha_{1},0 + \alpha_{2,1}n \dots \alpha_{1,m_{2}-1}n^{m_{2}-1})r_{2}^{n} +\dots + (\alpha_{t},0 + \alpha_{t,1}n \dots \alpha_{t,m_{t}-1}n^{m_{t}-1})r_{t}^{n} $$<br />
<br />
for $n = 0, 1, 2,\dots,$ where $\alpha_{i,j}$ are constants for $1 \leq i \leq t\:\text{and}\: 0 \leq j \leq m_{i} - 1.$Combinatoryhttps://gateoverflow.in/339231/kenneth-rosen-edition-7th-exercise-8-question-51-page-no-527Wed, 06 May 2020 04:14:41 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 53 (Page No. 527)
https://gateoverflow.in/339230/kenneth-rosen-edition-7th-exercise-8-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.]Combinatoryhttps://gateoverflow.in/339230/kenneth-rosen-edition-7th-exercise-8-question-53-page-no-527Wed, 06 May 2020 03:48:44 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 50 (Page No. 527)
https://gateoverflow.in/339228/kenneth-rosen-edition-7th-exercise-8-question-50-page-no-527
<p>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 $n$ elements in random order, satisfies the recurrence relation</p>
<p>$$C_{n} = 1 + n + \dfrac{2}{n}\sum_{k=0}^{n-1}C_{k}$$
<br>
for $n = 1, 2, \dots,$ with initial condition $C_{0} = 0.$</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Show that $\{C_{n}\}$ also satisfies the recurrence relation $nC_{n} = (n + 1)C_{n-1} + 2n \:\text{for}\: n = 1, 2, \dots$</li>
<li>Use question $48$ to solve the recurrence relation in part $(A)$ to find an explicit formula for $C_{n}.$</li>
</ol>Combinatoryhttps://gateoverflow.in/339228/kenneth-rosen-edition-7th-exercise-8-question-50-page-no-527Wed, 06 May 2020 03:35:18 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 49 (Page No. 527)
https://gateoverflow.in/339224/kenneth-rosen-edition-7th-exercise-8-question-49-page-no-527
Use question $48$ to solve the recurrence relation $(n + 1)a_{n} = (n + 3)a_{n-1} + n, \:\text{for}\: n \geq 1, \:\text{with}\: a_{0} = 1$Combinatoryhttps://gateoverflow.in/339224/kenneth-rosen-edition-7th-exercise-8-question-49-page-no-527Wed, 06 May 2020 02:55:39 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 48 (Page No. 526)
https://gateoverflow.in/339223/kenneth-rosen-edition-7th-exercise-8-question-48-page-no-526
<p>Some linear recurrence relations that do not have constant coefficients can be systematically solved. This is the case for recurrence relations of the form $f (n)a_{n} = g(n)a_{n-1} + h(n).$ Exercises $48–50$ illustrate this.</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Show that the recurrence relation $f (n)a_{n} = g(n)a_{n-1} + h(n),$ for $n \geq 1,$ and with $a_{0} = C,$ can be reduced to a recurrence relation of the form $b_{n} = b_{n-1} + Q(n)h(n),$ where $b_{n} = g(n + 1)Q(n + 1)a_{n},$ with $Q(n) = \dfrac{(f (1)f (2) \dots f (n - 1))}{(g(1)g(2) \dots g(n))}.$</li>
<li>Use part $(A)$ to solve the original recurrence relation to obtain $a_{n} = \dfrac{C +\displaystyle{} \sum_{i = 1}^{n}Q(i)h(i)}{g(n + 1)Q(n + 1)}$</li>
</ol>Combinatoryhttps://gateoverflow.in/339223/kenneth-rosen-edition-7th-exercise-8-question-48-page-no-526Wed, 06 May 2020 02:50:24 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 47 (Page No. 526)
https://gateoverflow.in/339221/kenneth-rosen-edition-7th-exercise-8-question-47-page-no-526
<p>A new employee at an exciting new software company starts with a salary of $\$50,000$ and is promised that at the end of each year her salary will be double her salary of the previous year, with an extra increment of $\$10,000$ for each year she has been with the company.</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Construct a recurrence relation for her salary for her $n^{\text{th}}$ year of employment.</li>
<li>Solve this recurrence relation to find her salary for her $n^{\text{th}}$ year of employment.</li>
</ol>Combinatoryhttps://gateoverflow.in/339221/kenneth-rosen-edition-7th-exercise-8-question-47-page-no-526Wed, 06 May 2020 02:36:20 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 46 (Page No. 526)
https://gateoverflow.in/339220/kenneth-rosen-edition-7th-exercise-8-question-46-page-no-526
<p>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.</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Construct a recurrence relation for the number of goats on the island at the start of the $n^{\text{th}}$ year, assuming that during each year an extra $100$ goats are put on the island.</li>
<li>Solve the recurrence relation from part $(A)$ to find the number of goats on the island at the start of the $n^{th}$ year.</li>
<li>Construct a recurrence relation for the number of goats on the island at the start of the $n^{\text{th}}$ year, assuming that n goats are removed during the $n^{\text{th}}$ year for each $n \geq 3.$</li>
</ol>Combinatoryhttps://gateoverflow.in/339220/kenneth-rosen-edition-7th-exercise-8-question-46-page-no-526Wed, 06 May 2020 02:32:49 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 45 (Page No. 526)
https://gateoverflow.in/339218/kenneth-rosen-edition-7th-exercise-8-question-45-page-no-526
<p>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. None of the rabbits ever die or leave the island.</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Find a recurrence relation for the number of pairs of rabbits on the island $n$ months after one newborn pair is left on the island.</li>
<li>By solving the recurrence relation in $(A)$ determine the number of pairs of rabbits on the island $n$ months after one pair is left on the island.</li>
</ol>Combinatoryhttps://gateoverflow.in/339218/kenneth-rosen-edition-7th-exercise-8-question-45-page-no-526Wed, 06 May 2020 02:27:49 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 44 (Page No. 526)
https://gateoverflow.in/339217/kenneth-rosen-edition-7th-exercise-8-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 $d_{n}.$Combinatoryhttps://gateoverflow.in/339217/kenneth-rosen-edition-7th-exercise-8-question-44-page-no-526Wed, 06 May 2020 02:22:16 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 43 (Page No. 526)
https://gateoverflow.in/339216/kenneth-rosen-edition-7th-exercise-8-question-43-page-no-526
Express the solution of the linear nonhomogenous recurrence relation $a_{n} = a_{n-1} + a_{n-2} + 1\:\text{for}\: n \geq 2<br />
\:\text{where}\: a_{0} = 0\:\text{and}\: a_{1} = 1$ in terms of the Fibonacci numbers. [Hint: Let $b_{n} = a_{n + 1}$ and apply question $42$ to the sequence $b_{n}.]$Combinatoryhttps://gateoverflow.in/339216/kenneth-rosen-edition-7th-exercise-8-question-43-page-no-526Wed, 06 May 2020 02:18:52 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 42 (Page No. 526)
https://gateoverflow.in/339215/kenneth-rosen-edition-7th-exercise-8-question-42-page-no-526
Show that if $a_{n} = a_{n-1} + a_{n-2}, a_{0} = s\:\text{and}\: a_{1} = t,$ where $s$ and $t$ are constants, then $a_{n} = sf_{n-1} + tf_{n}$ for all positive integers $n.$Combinatoryhttps://gateoverflow.in/339215/kenneth-rosen-edition-7th-exercise-8-question-42-page-no-526Wed, 06 May 2020 02:13:43 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 41 (Page No. 526)
https://gateoverflow.in/339214/kenneth-rosen-edition-7th-exercise-8-question-41-page-no-526
<ol start="1" style="list-style-type:upper-alpha">
<li>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 $$\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}$$</li>
<li>Determine for which $n\: f_{n}$ is greater than $$\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}$$ and for which $n\: f_{n}$ is less than $$\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}.$$</li>
</ol>Combinatoryhttps://gateoverflow.in/339214/kenneth-rosen-edition-7th-exercise-8-question-41-page-no-526Wed, 06 May 2020 02:09:54 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 39 (Page No. 526)
https://gateoverflow.in/339207/kenneth-rosen-edition-7th-exercise-8-question-39-page-no-526
<ol start="1" style="list-style-type:upper-alpha">
<li>a) Find the characteristic roots of the linear homogeneous recurrence relation $a_{n} = a_{n-4}.$ [Note: These include complex numbers.]</li>
<li>Find the solution of the recurrence relation in part $(A)$ with $a_{0} = 1, a_{1} = 0, a_{2} = -1,\: \text{and}\: a_{3} = 1.$
<br>
</li>
</ol>Combinatoryhttps://gateoverflow.in/339207/kenneth-rosen-edition-7th-exercise-8-question-39-page-no-526Tue, 05 May 2020 16:46:44 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 37 (Page No. 526)
https://gateoverflow.in/339205/kenneth-rosen-edition-7th-exercise-8-question-37-page-no-526
Let an be the sum of the first $n$ triangular numbers, that is,<br />
$a_{n} = \displaystyle{}\sum_{k = 1}^{n} t_{k},\:\text{where}\: t_{k} = k(k + 1)/2.$ Show that $\{an\}$ satisfies the linear nonhomogeneous recurrence relation $a_{n} = a_{n-1} + n(n + 1)/2$ and the initial condition $a_{1} = 1.$<br />
Use Theorem $6$ to determine a formula for $a_{n}$ by solving this recurrence relation.Combinatoryhttps://gateoverflow.in/339205/kenneth-rosen-edition-7th-exercise-8-question-37-page-no-526Tue, 05 May 2020 16:33:31 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 36 (Page No. 526)
https://gateoverflow.in/339178/kenneth-rosen-edition-7th-exercise-8-question-36-page-no-526
Let an be the sum of the first $n$ perfect squares, that is, $a_{n} = \displaystyle{}\sum_{k = 1}^{n} k^{2}.$ Show that the sequence $\{a_{n}\}$ satisfies the linear nonhomogeneous recurrence relation $a_{n} = a_{n-1} + n^{2}$ and the initial condition $a_{1} = 1.$ Use<br />
Theorem $6$ to determine a formula for $a_{n}$ by solving this recurrence relation.Combinatoryhttps://gateoverflow.in/339178/kenneth-rosen-edition-7th-exercise-8-question-36-page-no-526Tue, 05 May 2020 04:49:36 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 35 (Page No. 526)
https://gateoverflow.in/339177/kenneth-rosen-edition-7th-exercise-8-question-35-page-no-526
Find the solution of the recurrence relation $a_{n} = 4a_{n-1} - 3a_{n-2} + 2^{n} + n + 3\:\text{with}\: a_{0} = 1\:\text{and}\: a_{1} = 4.$Combinatoryhttps://gateoverflow.in/339177/kenneth-rosen-edition-7th-exercise-8-question-35-page-no-526Tue, 05 May 2020 04:45:29 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 34 (Page No. 526)
https://gateoverflow.in/339175/kenneth-rosen-edition-7th-exercise-8-question-34-page-no-526
Find all solutions of the recurrence relation $a_{n} =7a_{n-1} - 16a_{n-2} + 12a_{n-3} + n4^{n}\:\text{with}\: a_{0} = -2,a_{1} = 0,\:\text{and}\: a_{2} = 5.$Combinatoryhttps://gateoverflow.in/339175/kenneth-rosen-edition-7th-exercise-8-question-34-page-no-526Tue, 05 May 2020 04:43:05 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 33 (Page No. 525)
https://gateoverflow.in/339174/kenneth-rosen-edition-7th-exercise-8-question-33-page-no-525
Find all solutions of the recurrence relation $a_{n} = 4a_{n-1} - 4a_{n-2} + (n + 1)2^{n}.$Combinatoryhttps://gateoverflow.in/339174/kenneth-rosen-edition-7th-exercise-8-question-33-page-no-525Tue, 05 May 2020 04:40:31 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 32 (Page No. 525)
https://gateoverflow.in/339173/kenneth-rosen-edition-7th-exercise-8-question-32-page-no-525
Find the solution of the recurrence relation $a_{n} = 2a_{n-1} + 3 \cdot 2^{n}.$Combinatoryhttps://gateoverflow.in/339173/kenneth-rosen-edition-7th-exercise-8-question-32-page-no-525Tue, 05 May 2020 04:35:01 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 31 (Page No. 525)
https://gateoverflow.in/339172/kenneth-rosen-edition-7th-exercise-8-question-31-page-no-525
Find all solutions of the recurrence relation $a_{n} = 5a_{n-1} - 6a_{n-2} + 2^{n}+ 3n.$ [Hint: Look for a particular solution of the form $qn2^{n} + p_{1}n + p_{2},$ where $q, p_{1}, \text{and}\: p_{2}$ are constants.]Combinatoryhttps://gateoverflow.in/339172/kenneth-rosen-edition-7th-exercise-8-question-31-page-no-525Tue, 05 May 2020 04:32:53 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 30 (Page No. 525)
https://gateoverflow.in/339171/kenneth-rosen-edition-7th-exercise-8-question-30-page-no-525
<ol start="1" style="list-style-type:upper-alpha">
<li>Find all solutions of the recurrence relation $a_{n} = -5a_{n-1} - 6a_{n-2} + 42 \cdot 4^{n}.$</li>
<li>Find the solution of this recurrence relation with $a_{1} = 56\:\text{and}\: a_{2} = 278.$
<br>
</li>
</ol>Combinatoryhttps://gateoverflow.in/339171/kenneth-rosen-edition-7th-exercise-8-question-30-page-no-525Tue, 05 May 2020 04:23:46 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 29 (Page No. 525)
https://gateoverflow.in/339170/kenneth-rosen-edition-7th-exercise-8-question-29-page-no-525
<ol start="1" style="list-style-type:upper-alpha">
<li>Find all solutions of the recurrence relation $a_{n} = 2a_{n-1} + 3n.$</li>
<li>Find the solution of the recurrence relation in part $(A)$ with initial condition $a_{1} = 5.$
<br>
</li>
</ol>Combinatoryhttps://gateoverflow.in/339170/kenneth-rosen-edition-7th-exercise-8-question-29-page-no-525Tue, 05 May 2020 04:18:30 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 28 (Page No. 525)
https://gateoverflow.in/339169/kenneth-rosen-edition-7th-exercise-8-question-28-page-no-525
<ol start="1" style="list-style-type:upper-alpha">
<li>Find all solutions of the recurrence relation $a_{n} = 2a_{n-1} + 2n^{2}.$</li>
<li>Find the solution of the recurrence relation in part $(A)$ with initial condition $a_{1} = 4.$
<br>
</li>
</ol>Combinatoryhttps://gateoverflow.in/339169/kenneth-rosen-edition-7th-exercise-8-question-28-page-no-525Tue, 05 May 2020 04:16:09 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 27 (Page No. 525)
https://gateoverflow.in/339168/kenneth-rosen-edition-7th-exercise-8-question-27-page-no-525
<p>What is the general form of the particular solution guaranteed to exist by Theorem 6 of the linear nonhomogeneous recurrence relation $a_{n} = 8a_{n-2} - 16a_{n-4} + F(n)$ if</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$F(n) = n^{3}?$</li>
<li>$F(n) = (-2)^{n}?$</li>
<li>$F(n) = n2^{n}? $</li>
<li>$F(n) = n^{2}4^{n}?$</li>
<li>$F(n) = (n^{2} - 2)(-2)^{n}?$</li>
<li>$F(n) = n^{4}2^{n}?$</li>
<li>$F(n) = 2?$
<br>
</li>
</ol>Combinatoryhttps://gateoverflow.in/339168/kenneth-rosen-edition-7th-exercise-8-question-27-page-no-525Tue, 05 May 2020 04:11:30 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 26 (Page No. 525)
https://gateoverflow.in/339167/kenneth-rosen-edition-7th-exercise-8-question-26-page-no-525
<p>What is the general form of the particular solution guaranteed to exist by Theorem $6$ of the linear nonhomogeneous recurrence relation $a_{n} = 6a_{n-1} - 12a_{n-2} + 8a_{n-3} + F (n)$ if</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$F (n) = n^{2}?$</li>
<li>$F (n) = 2^{n}?$</li>
<li>$F (n) = n2^{n}?$</li>
<li>$F (n) = (-2)^{n}?$</li>
<li>$F (n) = n^{2}2^{n}?$</li>
<li>$F (n) = n^{3}(-2)^{n}?$</li>
<li>$F (n) = 3?$
<br>
</li>
</ol>Combinatoryhttps://gateoverflow.in/339167/kenneth-rosen-edition-7th-exercise-8-question-26-page-no-525Tue, 05 May 2020 04:04:13 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 25 (Page No. 525)
https://gateoverflow.in/339166/kenneth-rosen-edition-7th-exercise-8-question-25-page-no-525
<ol start="1" style="list-style-type:upper-alpha">
<li>Determine values of the constants $A$ and $B$ such that $a_{n} = A{n} + B$ is a solution of recurrence relation $a_{n} = 2a_{n-1} + n + 5.$</li>
<li>Use Theorem $5$ to find all solutions of this recurrence relation.</li>
<li>Find the solution of this recurrence relation with $a_{0} = 4.$</li>
</ol>Combinatoryhttps://gateoverflow.in/339166/kenneth-rosen-edition-7th-exercise-8-question-25-page-no-525Tue, 05 May 2020 03:58:01 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 24 (Page No. 525)
https://gateoverflow.in/339165/kenneth-rosen-edition-7th-exercise-8-question-24-page-no-525
<p>Consider the nonhomogeneous linear recurrence relation $a_{n} = 2a_{n-1} + 2^{n}.$</p>
<ol>
<li>Show that $a_{n} = n2^{n}$ is a solution of this recurrence relation.</li>
<li>Use Theorem $5$ to find all solutions of this recurrence relation.</li>
<li>Find the solution with $a_{0} = 2.$
<br>
</li>
</ol>Combinatoryhttps://gateoverflow.in/339165/kenneth-rosen-edition-7th-exercise-8-question-24-page-no-525Tue, 05 May 2020 03:51:37 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 23 (Page No. 525)
https://gateoverflow.in/339164/kenneth-rosen-edition-7th-exercise-8-question-23-page-no-525
<p>Consider the nonhomogeneous linear recurrence relation $a_{n} = 3a_{n-1} + 2^{n}.$</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Show that $a_{n} = -2^{n+1}$ is a solution of this recurrence relation.</li>
<li>Use Theorem $5$ to find all solutions of this recurrence relation.</li>
<li>Find the solution with $a_{0} = 1.$
<br>
</li>
</ol>Combinatoryhttps://gateoverflow.in/339164/kenneth-rosen-edition-7th-exercise-8-question-23-page-no-525Tue, 05 May 2020 03:47:49 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 17 (Page No. 525)
https://gateoverflow.in/339102/kenneth-rosen-edition-7th-exercise-8-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 $k = n/2 .$ [Hint: Let $a_{n} = C(n, 0) + C(n − 1, 1) +\dots·+ C(n − k, k).$ Show that the sequence $\{a_{n}\}$ satisfies the same recurrence relation and initial conditions satisfied by the sequence of Fibonacci numbers.]Combinatoryhttps://gateoverflow.in/339102/kenneth-rosen-edition-7th-exercise-8-question-17-page-no-525Sun, 03 May 2020 16:14:36 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 16 (Page No. 525)
https://gateoverflow.in/339100/kenneth-rosen-edition-7th-exercise-8-question-16-page-no-525
Prove Theorem $3:$<br />
Let $c_{1},c_{2},\dots,c_{k}$ be real numbers. Suppose that the characteristic equation $$r^{k}-c_{1}r^{k-1}-\dots – c_{k} = 0$$<br />
has $k$ distinct roots $r_{1},r_{2},\dots r_{k}.$ Then a sequence $\{a_{n}\}$ is a solution of the recurrence relation $$a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k}$$ if and only if<br />
$$a_{n} = \alpha_{1}r^{n}_{1} + \alpha_{2}r_{2}^{n} + \dots + \alpha_{k}r^{n}_{k}$$<br />
for $n = 0,1,2,\dots,$ where $\alpha_{1},\alpha_{2},\dots,\alpha_{k}$ are constants.Combinatoryhttps://gateoverflow.in/339100/kenneth-rosen-edition-7th-exercise-8-question-16-page-no-525Sun, 03 May 2020 16:10:50 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 11 (Page No. 525)
https://gateoverflow.in/339082/kenneth-rosen-edition-7th-exercise-8-question-11-page-no-525
<p>The Lucas numbers satisfy the recurrence relation $L_{n} = L_{n−1} + L_{n−2},$ and the initial conditions $L_{0} = 2$ and $L_{1} = 1.$ </p>
<ol start="1" style="list-style-type:upper-alpha">
<li>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.</li>
<li>Find an explicit formula for the Lucas numbers.</li>
</ol>Combinatoryhttps://gateoverflow.in/339082/kenneth-rosen-edition-7th-exercise-8-question-11-page-no-525Sun, 03 May 2020 10:36:32 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 10 (Page No. 525)
https://gateoverflow.in/339081/kenneth-rosen-edition-7th-exercise-8-question-10-page-no-525
Prove Theorem $2:$ Let $c_{1}$ and $c_{2}$ be real numbers with $c_{2}\neq 0.$ Suppose that $r^{2}-c_{1}r-c_{2} = 0$ has only one root $r_{0}.$ A sequence $\{a_{n}\}$ is a solution of the recurrence relation $a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2}$ if and only if $a_{n} = \alpha_{1}r_{0}^{n} + \alpha_{2}nr_{0}^{n},$ for $n = 0,1,2,\dots,$ where $\alpha_{1}$ and $\alpha_{2}$ are constants.Combinatoryhttps://gateoverflow.in/339081/kenneth-rosen-edition-7th-exercise-8-question-10-page-no-525Sun, 03 May 2020 10:32:38 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 9 (Page No. 525)
https://gateoverflow.in/339079/kenneth-rosen-edition-7th-exercise-8-2-question-9-page-no-525
<p>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 $20\%$ of the amount in the account during that year. The second dividend is $45\%$ of the amount in the account in the previous year.</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Find a recurrence relation for $\{Pn\},$ where $P_{n}$ is the amount in the account at the end of $n$ years if no money is ever withdrawn.</li>
<li>How much is in the account after $n$ years if no money has been withdrawn?</li>
</ol>Combinatoryhttps://gateoverflow.in/339079/kenneth-rosen-edition-7th-exercise-8-2-question-9-page-no-525Sun, 03 May 2020 10:22:58 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 8 (Page No. 524 - 525)
https://gateoverflow.in/339077/kenneth-rosen-edition-7th-exercise-question-page-no-524-525
<p>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.</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Find a recurrence relation for $\{L_{n}\},$ where $L_{n}$ is the number of lobsters caught in year $n,$ under the assumption for this model.</li>
<li>Find $L_{n}$ if $100,000$ lobsters were caught in year $1\:\text{ and}\: 300,000$ were caught in year $2.$</li>
</ol>Combinatoryhttps://gateoverflow.in/339077/kenneth-rosen-edition-7th-exercise-question-page-no-524-525Sun, 03 May 2020 10:20:46 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 7 (Page No. 524)
https://gateoverflow.in/339076/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?Combinatoryhttps://gateoverflow.in/339076/kenneth-rosen-edition-7th-exercise-8-2-question-7-page-no-524Sun, 03 May 2020 10:18:28 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 6 (Page No. 524)
https://gateoverflow.in/339075/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?Combinatoryhttps://gateoverflow.in/339075/kenneth-rosen-edition-7th-exercise-8-2-question-6-page-no-524Sun, 03 May 2020 10:17:07 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 5 (Page No. 524)
https://gateoverflow.in/339074/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 $8.1?$Combinatoryhttps://gateoverflow.in/339074/kenneth-rosen-edition-7th-exercise-8-2-question-5-page-no-524Sun, 03 May 2020 10:15:48 +0000Kenneth Rosen Edition 7th Exercise 8.2 Question 4 (Page No. 524)
https://gateoverflow.in/339073/kenneth-rosen-edition-7th-exercise-8-2-question-4-page-no-524
<p>Solve these recurrence relations together with the initial conditions given.</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$a_{n} = a_{n-1}+ 6a_{n-2} \:\text{for}\: n \geq 2, a_{0} = 3, a_{1} = 6$</li>
<li>$a_{n} = 7a_{n-1}− 10a_{n-2} \:\text{for}\: n \geq 2, a_{0} = 2, a_{1} = 1$</li>
<li>$a_{n} = 6a_{n-1}− 8a_{n-2} \:\text{for}\: n \geq 2, a_{0} = 4, a_{1} = 10$</li>
<li>$a_{n} = 2a_{n-1}− a_{n-2} \:\text{for}\: n \geq 2, a_{0} = 4, a_{1} = 1$</li>
<li>$a_{n} = a_{n-2} \:\text{for}\: n \geq 2, a_{0} = 5, a_{1} = −1$</li>
<li>$a_{n} = −6a_{n-1}− 9a_{n-2} \:\text{for}\: n \geq 2, a_{0} = 3, a_{1} = −3$</li>
<li>$a_{n+2} = −4a_{n+1} + 5a_{n} \:\text{for}\: n \geq 0, a_{0} = 2, a_{1} = 8$</li>
</ol>Combinatoryhttps://gateoverflow.in/339073/kenneth-rosen-edition-7th-exercise-8-2-question-4-page-no-524Sun, 03 May 2020 10:12:40 +0000Kenneth Rosen Edition 7th Exercise 8.1 Question 57 (Page No. 512)
https://gateoverflow.in/339060/kenneth-rosen-edition-7th-exercise-8-question-57-page-no-512
<p>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 using the fewest integer multiplications, where $A_{1}, A_{2},\dots, A_{n}\:\text{are}\: m_{1} \times m_{2}, m_{2} \times m_{3},\dots,m_{n} \times m_{n+1}$ matrices, respectively, and each matrix has integer entries. Recall that by the associative law, the product does not depend on the order in which the matrices are multiplied.</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Show that the brute-force method of determining the minimum number of integer multiplications needed to solve a matrix-chain multiplication problem has exponential worst-case complexity. [Hint: Do this by first showing that the order of multiplication of matrices is specified by parenthesizing the product. Then, use Example $5$ and the result of part $(A)$ of question $41$ in Section $8.4.]$</li>
<li>Denote by $A_{ij}$ the product $A_{i}A_{i+1}\dots, A_{j},$ and $M(i,j)$ the minimum number of integer multiplications required to find $A_{ij}.$ Show that if the least number of integer multiplications are used to compute $A_{ij},$ where $i<j,$ by splitting the product into the product of $A_{i}$ through $A_{k}$ and the product of $A_{k+1}$ through $A_{j},$ then the first $k$ terms must be parenthesized so that $A_{ik}$ is computed in the optimal way using $M(i,k)$ integer multiplications and $A_{{k+1},j}$ must be parenthesized so that $A_{{k+1},j}$ is computed in the optimal way using $M(k+1,j)$ integer multiplications.</li>
<li>Explain why part $(B)$ leads to the recurrence relation $M(i,j)=\text{min}_{i\leq k < j}(M(i,k) + M(k+1,j) + m_{i}m_{k+1}m_{j+1})\:\text{if}\:1\leq i \leq j < j\leq n.$</li>
<li>Use the recurrence relation in part $(C)$ to construct an efficient algorithm for determining the order the $n$ matrices should be multiplied to use the minimum number of integer multiplications. Store the partial results $M(i, j )$ as you find them so that your algorithm will not have exponential complexity.</li>
<li>Show that your algorithm from part $(D)$ has $O(n^{3})$ worst-case complexity in terms of multiplications of integers.</li>
</ol>Combinatoryhttps://gateoverflow.in/339060/kenneth-rosen-edition-7th-exercise-8-question-57-page-no-512Sun, 03 May 2020 07:02:35 +0000Kenneth Rosen Edition 7th Exercise 8.1 Question 56 (Page No. 512)
https://gateoverflow.in/339057/kenneth-rosen-edition-7th-exercise-8-question-56-page-no-512
<p>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 $a_{1}, a_{2},\dots,a_{n},$ the algorithm computes the maximum sum $\displaystyle\sum_{i=j}^{k} a_{i}\: \text{where}\: 1 \leq j \leq k \leq n.$</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>Show that if all terms of the sequence are nonnegative, this problem is solved by taking the sum of all terms. Then, give an example where the maximum sum of consecutive terms is not the sum of all terms.</li>
<li>Let $M(k)$ be the maximum of the sums of consecutive terms of the sequence ending at $a_{k}.$ That is, $M(k) = \:\text{max}\:_{1\leq j\leq k}\displaystyle{} \sum_{i=j}^{k} a_{i}.$ Explain why the recurrence relation $M(k) = \text{max}(M(k − 1) + a_{k}, a_{k})$ holds for $k = 2, \dots, n.$</li>
<li>Use part $(B)$ to develop a dynamic programming algorithm for solving this problem.</li>
<li>Show each step your algorithm from part $(C)$ uses to find the maximum sum of consecutive terms of the sequence $2, −3, 4, 1, −2, 3.$ </li>
<li>Show that the worst-case complexity in terms of the number of additions and comparisons of your algorithm from part $(C)$ is linear.</li>
</ol>Combinatoryhttps://gateoverflow.in/339057/kenneth-rosen-edition-7th-exercise-8-question-56-page-no-512Sun, 03 May 2020 06:43:02 +0000Kenneth Rosen Edition 7th Exercise 8.1 Question 55 (Page No. 512)
https://gateoverflow.in/339048/kenneth-rosen-edition-7th-exercise-8-question-55-page-no-512
<p>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.</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$20, 10, 50, 30, 15, 25, 40.$ </li>
<li>$100, 5, 10, 20, 25, 40, 30. $</li>
<li>$2, 3, 8, 5, 4, 7, 10. $</li>
<li>$10, 8, 7, 25, 20, 30, 5.$</li>
</ol>Combinatoryhttps://gateoverflow.in/339048/kenneth-rosen-edition-7th-exercise-8-question-55-page-no-512Sun, 03 May 2020 05:03:33 +0000Kenneth Rosen Edition 7th Exercise 8.1 Question 54 (Page No. 512)
https://gateoverflow.in/339047/kenneth-rosen-edition-7th-exercise-8-question-54-page-no-512
<p>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</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$20, 10, 50, 30, 15, 25, 40.$ </li>
<li>$100, 5, 10, 20, 25, 40, 30. $</li>
<li>$2, 3, 8, 5, 4, 7, 10. $</li>
<li>$10, 8, 7, 25, 20, 30, 5.$</li>
</ol>Combinatoryhttps://gateoverflow.in/339047/kenneth-rosen-edition-7th-exercise-8-question-54-page-no-512Sun, 03 May 2020 05:02:17 +0000Kenneth Rosen Edition 7th Exercise 8.1 Question 53 (Page No. 512)
https://gateoverflow.in/339046/kenneth-rosen-edition-7th-exercise-8-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 $1.$Combinatoryhttps://gateoverflow.in/339046/kenneth-rosen-edition-7th-exercise-8-question-53-page-no-512Sun, 03 May 2020 05:00:30 +0000Kenneth Rosen Edition 7th Exercise 8.1 Question 52 (Page No. 512)
https://gateoverflow.in/339044/kenneth-rosen-edition-7th-exercise-8-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 $\triangledown a_{n}$ is<br />
$$\triangledown a_{n} = a_{n} − a_{n−1}.$$<br />
The $(k + 1)^{\text{st}}$ difference $\triangledown^{k+1}a_{n}$ is obtained from $\triangledown ^{k} a_{n}$ by<br />
$$\triangledown ^{k+1}a_{n} = \triangledown^{k}a_{n} − \triangledown ^{k}a_{n−1}.$$<br />
Show that any recurrence relation for the sequence $\{a_{n}\}$ can be written in terms of $a_{n}, \triangledown a_{n}, \triangledown^{2}a_{n},\dots$ The resulting equation involving the sequences and its differences is called a difference equation.Combinatoryhttps://gateoverflow.in/339044/kenneth-rosen-edition-7th-exercise-8-question-52-page-no-512Sun, 03 May 2020 04:52:27 +0000