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Kenneth Rosen Edition 7 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 $C_{n}.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 49 (Page No. 527)
Use question $48$ to solve the recurrence relation $(n + 1)a_{n} = (n + 3)a_{n-1} + n, \:\text{for}\: n \geq 1, \:\text{with}\: a_{0} = 1$
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Kenneth Rosen Edition 7 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 $f (n)a_{n} = g(n)a_{n-1} + h(n).$ Exercises $48-50$ ...
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Kenneth Rosen Edition 7 Exercise 8.2 Question 47 (Page No. 526)
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 $\ ... year of employment. Solve this recurrence relation to find her salary for her $n^{\text{th}}$ year of employment.
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Kenneth Rosen Edition 7 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 on ... assuming that n goats are removed during the $n^{\text{th}}$ year for each $n \geq 3.$
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Kenneth Rosen Edition 7 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.
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Kenneth Rosen Edition 7 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 $d_{n}.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 43 (Page No. 526)
Express the solution of the linear nonhomogenous recurrence relation $a_{n} = a_{n-1} + a_{n-2} + 1\:\text{for}\: n \geq 2 \:\text{where}\: a_{0} = 0\:\text{and}\: a_{1} = 1$ in terms of the Fibonacci numbers. [Hint: Let $b_{n} = a_{n + 1}$ and apply question $42$ to the sequence $b_{n}.]$
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Kenneth Rosen Edition 7 Exercise 8.2 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 $n.$
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Kenneth Rosen Edition 7 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 greater ... for which $n\: f_{n}$ is less than $\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 40 (Page No. 526)
Solve the simultaneous recurrence relations $a_{n} = 3a_{n-1} + 2b_{n-1}$ $b_{n} = a_{n-1} + 2b_{n-1}$ with $a_{0} = 1 \: \text{and}\: b_{0} = 2.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 39 (Page No. 526)
a) Find the characteristic roots of the linear homogeneous recurrence relation $a_{n} = a_{n-4}.$ [Note: These include complex numbers.] Find the solution of the recurrence relation in part $(A)$ with $a_{0} = 1, a_{1} = 0, a_{2} = -1,\: \text{and}\: a_{3} = 1.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 38 (Page No. 526)
Find the characteristic roots of the linear homogeneous recurrence relation $a_{n} = 2a_{n-1} - 2a_{n-2}.$ [Note: These are complex numbers.] Find the solution of the recurrence relation in part $(A)$ with $a_{0} = 1\:\text{and}\: a_{1} = 2.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 37 (Page No. 526)
Let an be the sum of the first $n$ triangular numbers, that is, $a_{n} = \displaystyle{}\sum_{k = 1}^{n} t_{k},\:\text{where}\: t_{k} = k(k + 1)/2.$ Show that $\{an\}$ satisfies the linear nonhomogeneous ... and the initial condition $a_{1} = 1.$ Use Theorem $6$ to determine a formula for $a_{n}$ by solving this recurrence relation.
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Kenneth Rosen Edition 7 Exercise 8.2 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 $\{a_{n}\}$ ... initial condition $a_{1} = 1.$ Use Theorem $6$ to determine a formula for $a_{n}$ by solving this recurrence relation.
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Kenneth Rosen Edition 7 Exercise 8.2 Question 35 (Page No. 526)
Find the solution of the recurrence relation $a_{n} = 4a_{n-1} - 3a_{n-2} + 2^{n} + n + 3\:\text{with}\: a_{0} = 1\:\text{and}\: a_{1} = 4.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 34 (Page No. 526)
Find all solutions of the recurrence relation $a_{n} =7a_{n-1} - 16a_{n-2} + 12a_{n-3} + n4^{n}\:\text{with}\: a_{0} = -2,a_{1} = 0,\:\text{and}\: a_{2} = 5.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 33 (Page No. 525)
Find all solutions of the recurrence relation $a_{n} = 4a_{n-1} - 4a_{n-2} + (n + 1)2^{n}.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 32 (Page No. 525)
Find the solution of the recurrence relation $a_{n} = 2a_{n-1} + 3 \cdot 2^{n}.$
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Kenneth Rosen Edition 7 Exercise 8.2 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.]
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Kenneth Rosen Edition 7 Exercise 8.2 Question 30 (Page No. 525)
Find all solutions of the recurrence relation $a_{n} = -5a_{n-1} - 6a_{n-2} + 42 \cdot 4^{n}.$ Find the solution of this recurrence relation with $a_{1} = 56\:\text{and}\: a_{2} = 278.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 29 (Page No. 525)
Find all solutions of the recurrence relation $a_{n} = 2a_{n-1} + 3n.$ Find the solution of the recurrence relation in part $(A)$ with initial condition $a_{1} = 5.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 28 (Page No. 525)
Find all solutions of the recurrence relation $a_{n} = 2a_{n-1} + 2n^{2}.$ Find the solution of the recurrence relation in part $(A)$ with initial condition $a_{1} = 4.$
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Kenneth Rosen Edition 7 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 $a_{n} = 8a_{n-2} - 16a_{n-4} + F(n)$ 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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Kenneth Rosen Edition 7 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 $a_{n} = 6a_{n-1} - 12a_{n-2} + 8a_{n-3} + F (n)$ 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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Kenneth Rosen Edition 7 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 $a_{n} = 2a_{n-1} + n + 5.$ Use Theorem $5$ to find all solutions of this recurrence relation. Find the solution of this recurrence relation with $a_{0} = 4.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 24 (Page No. 525)
Consider the nonhomogeneous linear recurrence relation $a_{n} = 2a_{n-1} + 2^{n}.$ 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 $a_{0} = 2.$
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Kenneth Rosen Edition 7 Exercise 8.2 Question 23 (Page No. 525)
Consider the nonhomogeneous linear recurrence relation $a_{n} = 3a_{n-1} + 2^{n}.$ 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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Kenneth Rosen Edition 7 Exercise 8.2 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 $-1, -1, -1, 2, 2, 5, 5, 7?$
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Kenneth Rosen Edition 7 Exercise 8.2 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 $1,1,1,1,−2,−2,−2,3,3,−4?$
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