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GATE Overflow - Recent questions in Discrete Mathematics https://db.gateoverflow.in/questions/mathematics/discrete-mathematics Powered by Question2Answer Kenneth Rosen Edition 7th Exercise 8.2 Question 52 (Page No. 527) https://db.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 $$n^{m}(p_{t}n^{t} + p_{t-1}n^{t-1} + \dots + p_{1}n + p_{0})s^{n}.$$ Combinatory https://db.gateoverflow.in/339232/kenneth-rosen-edition-7th-exercise-8-question-52-page-no-527 Wed, 06 May 2020 10:03:22 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 51 (Page No. 527) https://db.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 /> $$a_{n} = (\alpha_{1},0 + \alpha_{1,1n} + &nbsp;\dots + \alpha_{1,m_{1}-1}n^{m_{1}-1})r_{1}^{n} + (\alpha_{1},0 + \alpha_{2,1}n &nbsp;\dots \alpha_{1,m_{2}-1}n^{m_{2}-1})r_{2}^{n} +\dots + (\alpha_{t},0 + \alpha_{t,1}n &nbsp;\dots \alpha_{t,m_{t}-1}n^{m_{t}-1})r_{t}^{n}$$<br /> <br /> for $n = 0, 1, 2,\dots,$ where $\alpha_{i,j}$ are constants for $1 \leq i \leq t\:\text{and}\: 0 \leq j \leq m_{i} - 1.$ Combinatory https://db.gateoverflow.in/339231/kenneth-rosen-edition-7th-exercise-8-question-51-page-no-527 Wed, 06 May 2020 09:44:41 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 53 (Page No. 527) https://db.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.] Combinatory https://db.gateoverflow.in/339230/kenneth-rosen-edition-7th-exercise-8-question-53-page-no-527 Wed, 06 May 2020 09:18:44 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 50 (Page No. 527) https://db.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> Combinatory https://db.gateoverflow.in/339228/kenneth-rosen-edition-7th-exercise-8-question-50-page-no-527 Wed, 06 May 2020 09:05:18 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 49 (Page No. 527) https://db.gateoverflow.in/339224/kenneth-rosen-edition-7th-exercise-8-question-49-page-no-527 Use question $48$ to solve the recurrence relation $(n + 1)a_{n} = (n + 3)a_{n-1} + n, \:\text{for}\: n \geq 1, \:\text{with}\: a_{0} = 1$ Combinatory https://db.gateoverflow.in/339224/kenneth-rosen-edition-7th-exercise-8-question-49-page-no-527 Wed, 06 May 2020 08:25:39 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 48 (Page No. 526) https://db.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> Combinatory https://db.gateoverflow.in/339223/kenneth-rosen-edition-7th-exercise-8-question-48-page-no-526 Wed, 06 May 2020 08:20:24 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 47 (Page No. 526) https://db.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> Combinatory https://db.gateoverflow.in/339221/kenneth-rosen-edition-7th-exercise-8-question-47-page-no-526 Wed, 06 May 2020 08:06:20 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 46 (Page No. 526) https://db.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> Combinatory https://db.gateoverflow.in/339220/kenneth-rosen-edition-7th-exercise-8-question-46-page-no-526 Wed, 06 May 2020 08:02:49 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 45 (Page No. 526) https://db.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> Combinatory https://db.gateoverflow.in/339218/kenneth-rosen-edition-7th-exercise-8-question-45-page-no-526 Wed, 06 May 2020 07:57:49 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 44 (Page No. 526) https://db.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 $d_{n}.$ Combinatory https://db.gateoverflow.in/339217/kenneth-rosen-edition-7th-exercise-8-question-44-page-no-526 Wed, 06 May 2020 07:52:16 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 43 (Page No. 526) https://db.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 $b_{n}.]$ Combinatory https://db.gateoverflow.in/339216/kenneth-rosen-edition-7th-exercise-8-question-43-page-no-526 Wed, 06 May 2020 07:48:52 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 42 (Page No. 526) https://db.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 $n.$ Combinatory https://db.gateoverflow.in/339215/kenneth-rosen-edition-7th-exercise-8-question-42-page-no-526 Wed, 06 May 2020 07:43:43 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 41 (Page No. 526) https://db.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> Combinatory https://db.gateoverflow.in/339214/kenneth-rosen-edition-7th-exercise-8-question-41-page-no-526 Wed, 06 May 2020 07:39:54 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 40 (Page No. 526) https://db.gateoverflow.in/339209/kenneth-rosen-edition-7th-exercise-8-question-40-page-no-526 <p>Solve the simultaneous recurrence relations</p> <ul> <li>$a_{n} = 3a_{n-1} + 2b_{n-1}$</li> <li>$b_{n} = a_{n-1} + 2b_{n-1}$</li> </ul> <p>with $a_{0} = 1 \: \text{and}\: b_{0} = 2.$</p> Combinatory https://db.gateoverflow.in/339209/kenneth-rosen-edition-7th-exercise-8-question-40-page-no-526 Tue, 05 May 2020 22:29:27 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 39 (Page No. 526) https://db.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> &nbsp;</li> </ol> Combinatory https://db.gateoverflow.in/339207/kenneth-rosen-edition-7th-exercise-8-question-39-page-no-526 Tue, 05 May 2020 22:16:44 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 38 (Page No. 526) https://db.gateoverflow.in/339206/kenneth-rosen-edition-7th-exercise-8-question-38-page-no-526 <ol start="1" style="list-style-type:upper-alpha"> <li>Find the characteristic roots of the linear homogeneous recurrence relation $a_{n} = 2a_{n-1} - 2a_{n-2}.$ [Note: These are complex numbers.]</li> <li>Find the solution of the recurrence relation in part $(A)$ with $a_{0} = 1\:\text{and}\: a_{1} = 2.$ <br> &nbsp;</li> </ol> Combinatory https://db.gateoverflow.in/339206/kenneth-rosen-edition-7th-exercise-8-question-38-page-no-526 Tue, 05 May 2020 22:11:06 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 37 (Page No. 526) https://db.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. Combinatory https://db.gateoverflow.in/339205/kenneth-rosen-edition-7th-exercise-8-question-37-page-no-526 Tue, 05 May 2020 22:03:31 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 36 (Page No. 526) https://db.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. Combinatory https://db.gateoverflow.in/339178/kenneth-rosen-edition-7th-exercise-8-question-36-page-no-526 Tue, 05 May 2020 10:19:36 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 35 (Page No. 526) https://db.gateoverflow.in/339177/kenneth-rosen-edition-7th-exercise-8-question-35-page-no-526 Find the solution of the recurrence relation $a_{n} = 4a_{n-1} - 3a_{n-2} + 2^{n} + n + 3\:\text{with}\: a_{0} = 1\:\text{and}\: a_{1} = 4.$ Combinatory https://db.gateoverflow.in/339177/kenneth-rosen-edition-7th-exercise-8-question-35-page-no-526 Tue, 05 May 2020 10:15:29 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 34 (Page No. 526) https://db.gateoverflow.in/339175/kenneth-rosen-edition-7th-exercise-8-question-34-page-no-526 Find all solutions of the recurrence relation $a_{n} =7a_{n-1} - 16a_{n-2} + 12a_{n-3} + n4^{n}\:\text{with}\: a_{0} = -2,a_{1} = 0,\:\text{and}\: a_{2} = 5.$ Combinatory https://db.gateoverflow.in/339175/kenneth-rosen-edition-7th-exercise-8-question-34-page-no-526 Tue, 05 May 2020 10:13:05 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 33 (Page No. 525) https://db.gateoverflow.in/339174/kenneth-rosen-edition-7th-exercise-8-question-33-page-no-525 Find all solutions of the recurrence relation $a_{n} = 4a_{n-1} - 4a_{n-2} + (n + 1)2^{n}.$ Combinatory https://db.gateoverflow.in/339174/kenneth-rosen-edition-7th-exercise-8-question-33-page-no-525 Tue, 05 May 2020 10:10:31 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 32 (Page No. 525) https://db.gateoverflow.in/339173/kenneth-rosen-edition-7th-exercise-8-question-32-page-no-525 Find the solution of the recurrence relation $a_{n} = 2a_{n-1} + 3 \cdot 2^{n}.$ Combinatory https://db.gateoverflow.in/339173/kenneth-rosen-edition-7th-exercise-8-question-32-page-no-525 Tue, 05 May 2020 10:05:01 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 31 (Page No. 525) https://db.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.] Combinatory https://db.gateoverflow.in/339172/kenneth-rosen-edition-7th-exercise-8-question-31-page-no-525 Tue, 05 May 2020 10:02:53 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 30 (Page No. 525) https://db.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> &nbsp;</li> </ol> Combinatory https://db.gateoverflow.in/339171/kenneth-rosen-edition-7th-exercise-8-question-30-page-no-525 Tue, 05 May 2020 09:53:46 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 29 (Page No. 525) https://db.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> &nbsp;</li> </ol> Combinatory https://db.gateoverflow.in/339170/kenneth-rosen-edition-7th-exercise-8-question-29-page-no-525 Tue, 05 May 2020 09:48:30 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 28 (Page No. 525) https://db.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> &nbsp;</li> </ol> Combinatory https://db.gateoverflow.in/339169/kenneth-rosen-edition-7th-exercise-8-question-28-page-no-525 Tue, 05 May 2020 09:46:09 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 27 (Page No. 525) https://db.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> &nbsp;</li> </ol> Combinatory https://db.gateoverflow.in/339168/kenneth-rosen-edition-7th-exercise-8-question-27-page-no-525 Tue, 05 May 2020 09:41:30 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 26 (Page No. 525) https://db.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> &nbsp;</li> </ol> Combinatory https://db.gateoverflow.in/339167/kenneth-rosen-edition-7th-exercise-8-question-26-page-no-525 Tue, 05 May 2020 09:34:13 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 25 (Page No. 525) https://db.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> Combinatory https://db.gateoverflow.in/339166/kenneth-rosen-edition-7th-exercise-8-question-25-page-no-525 Tue, 05 May 2020 09:28:01 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 24 (Page No. 525) https://db.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> &nbsp;</li> </ol> Combinatory https://db.gateoverflow.in/339165/kenneth-rosen-edition-7th-exercise-8-question-24-page-no-525 Tue, 05 May 2020 09:21:37 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 23 (Page No. 525) https://db.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> &nbsp;</li> </ol> Combinatory https://db.gateoverflow.in/339164/kenneth-rosen-edition-7th-exercise-8-question-23-page-no-525 Tue, 05 May 2020 09:17:49 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 22 (Page No. 525) https://db.gateoverflow.in/339163/kenneth-rosen-edition-7th-exercise-8-question-22-page-no-525 What is the general form of the solutions of a linear homogeneous recurrence relation if its characteristic equation has the roots $-1, -1, -1, 2, 2, 5, 5, 7?$ Combinatory https://db.gateoverflow.in/339163/kenneth-rosen-edition-7th-exercise-8-question-22-page-no-525 Tue, 05 May 2020 09:13:23 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 21 (Page No. 525) https://db.gateoverflow.in/339162/kenneth-rosen-edition-7th-exercise-8-question-21-page-no-525 What is the general form of the solutions of a linear homogeneous recurrence relation if its characteristic equation has roots $1,1,1,1,−2,−2,−2,3,3,−4?$ Combinatory https://db.gateoverflow.in/339162/kenneth-rosen-edition-7th-exercise-8-question-21-page-no-525 Tue, 05 May 2020 09:09:50 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 20 (Page No. 525) https://db.gateoverflow.in/339105/kenneth-rosen-edition-7th-exercise-8-question-20-page-no-525 Find the general form of the solutions of the recurrence relation $a_{n} = 8a_{n−2} − 16a_{n−4}.$ Combinatory https://db.gateoverflow.in/339105/kenneth-rosen-edition-7th-exercise-8-question-20-page-no-525 Sun, 03 May 2020 21:58:17 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 19 (Page No. 525) https://db.gateoverflow.in/339104/kenneth-rosen-edition-7th-exercise-8-question-19-page-no-525 Solve the recurrence relation $a_{n} = −3a_{n−1} − 3a_{n−2} − a_{n−3}\:\text{with}\: a_{0} = 5, a_{1} = −9,\:\text{and}\: a_{2} = 15.$ Combinatory https://db.gateoverflow.in/339104/kenneth-rosen-edition-7th-exercise-8-question-19-page-no-525 Sun, 03 May 2020 21:57:55 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 18 (Page No. 525) https://db.gateoverflow.in/339103/kenneth-rosen-edition-7th-exercise-8-question-18-page-no-525 Solve the recurrence relation $a_{n} = 6a_{n−1} − 12a_{n−2} + 8a_{n−3} \:\text{with}\: a_{0} = −5, a_{1} = 4,\: \text{and}\: a_{2} = 88.$ Combinatory https://db.gateoverflow.in/339103/kenneth-rosen-edition-7th-exercise-8-question-18-page-no-525 Sun, 03 May 2020 21:57:35 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 17 (Page No. 525) https://db.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.] Combinatory https://db.gateoverflow.in/339102/kenneth-rosen-edition-7th-exercise-8-question-17-page-no-525 Sun, 03 May 2020 21:44:36 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 16 (Page No. 525) https://db.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. Combinatory https://db.gateoverflow.in/339100/kenneth-rosen-edition-7th-exercise-8-question-16-page-no-525 Sun, 03 May 2020 21:40:50 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 15 (Page No. 525) https://db.gateoverflow.in/339086/kenneth-rosen-edition-7th-exercise-8-question-15-page-no-525 Find the solution to $a_{n} = 2a_{n−1} + 5a_{n−2} − 6a_{n−3}\: \text{with}\: a_{0} = 7, a_{1} = −4,\:\text{and}\: a_{2} = 8.$ Combinatory https://db.gateoverflow.in/339086/kenneth-rosen-edition-7th-exercise-8-question-15-page-no-525 Sun, 03 May 2020 16:26:39 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 14 (Page No. 525) https://db.gateoverflow.in/339085/kenneth-rosen-edition-7th-exercise-8-question-14-page-no-525 Find the solution to $a_{n} = 5a_{n−2}− 4a_{n−4} \:\text{with}\: a_{0} = 3, a_{1} = 2, a_{2} = 6, \:\text{and}\: a_{3} = 8.$ Combinatory https://db.gateoverflow.in/339085/kenneth-rosen-edition-7th-exercise-8-question-14-page-no-525 Sun, 03 May 2020 16:24:59 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 13 (Page No. 525) https://db.gateoverflow.in/339084/kenneth-rosen-edition-7th-exercise-8-question-13-page-no-525 Find the solution to $a_{n} = 7a_{n−2} + 6a_{n−3}\:\text{with}\: a_{0} = 9, a_{1} = 10, \text{and}\: a_{2} = 32.$ Combinatory https://db.gateoverflow.in/339084/kenneth-rosen-edition-7th-exercise-8-question-13-page-no-525 Sun, 03 May 2020 16:22:34 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 12 (Page No. 525) https://db.gateoverflow.in/339083/kenneth-rosen-edition-7th-exercise-8-question-12-page-no-525 Find the solution to $a_{n} = 2a_{n−1} + a_{n−2} − 2a_{n−3} \:\text{for}\: n = 3, 4, 5,\dots, \:\text{with}\: a_{0} = 3, a_{1} = 6, \:\text{and}\: a_{2} = 0.$ Combinatory https://db.gateoverflow.in/339083/kenneth-rosen-edition-7th-exercise-8-question-12-page-no-525 Sun, 03 May 2020 16:16:35 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 11 (Page No. 525) https://db.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 $L_{1} = 1.$&nbsp;</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> Combinatory https://db.gateoverflow.in/339082/kenneth-rosen-edition-7th-exercise-8-question-11-page-no-525 Sun, 03 May 2020 16:06:32 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 10 (Page No. 525) https://db.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. Combinatory https://db.gateoverflow.in/339081/kenneth-rosen-edition-7th-exercise-8-question-10-page-no-525 Sun, 03 May 2020 16:02:38 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 9 (Page No. 525) https://db.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> Combinatory https://db.gateoverflow.in/339079/kenneth-rosen-edition-7th-exercise-8-2-question-9-page-no-525 Sun, 03 May 2020 15:52:58 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 8 (Page No. 524 - 525) https://db.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> Combinatory https://db.gateoverflow.in/339077/kenneth-rosen-edition-7th-exercise-question-page-no-524-525 Sun, 03 May 2020 15:50:46 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 7 (Page No. 524) https://db.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? Combinatory https://db.gateoverflow.in/339076/kenneth-rosen-edition-7th-exercise-8-2-question-7-page-no-524 Sun, 03 May 2020 15:48:28 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 6 (Page No. 524) https://db.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? Combinatory https://db.gateoverflow.in/339075/kenneth-rosen-edition-7th-exercise-8-2-question-6-page-no-524 Sun, 03 May 2020 15:47:07 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 5 (Page No. 524) https://db.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$8.1?$Combinatory https://db.gateoverflow.in/339074/kenneth-rosen-edition-7th-exercise-8-2-question-5-page-no-524 Sun, 03 May 2020 15:45:48 +0000 Kenneth Rosen Edition 7th Exercise 8.2 Question 4 (Page No. 524) https://db.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> Combinatory https://db.gateoverflow.in/339073/kenneth-rosen-edition-7th-exercise-8-2-question-4-page-no-524 Sun, 03 May 2020 15:42:40 +0000