# Recent questions tagged kenneth-rosen

1
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 $a_{0} = 1, a_{1} = 0, a_{2} = -1,\: \text{and}\: a_{3} = 1.$
2
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 $a_{0} = 1\:\text{and}\: a_{1} = 2.$
3
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 recurrence relation $a_{n} = a_{n-1} + n(n + 1)/2$ and the initial condition $a_{1} = 1.$ Use Theorem $6$ to determine a formula for $a_{n}$ by solving this recurrence relation.
4
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 Theorem $6$ to determine a formula for $a_{n}$ by solving this recurrence relation.
5
Find the solution of the recurrence relation $a_{n} = 4a_{n-1} - 3a_{n-2} + 2^{n} + n + 3\:\text{with}\: a_{0} = 1\:\text{and}\: a_{1} = 4.$
6
Find all solutions of the recurrence relation $a_{n} =7a_{n-1} - 16a_{n-2} + 12a_{n-3} + n4^{n}\:\text{with}\: a_{0} = -2,a_{1} = 0,\:\text{and}\: a_{2} = 5.$
7
Find all solutions of the recurrence relation $a_{n} = 4a_{n-1} - 4a_{n-2} + (n + 1)2^{n}.$
8
Find the solution of the recurrence relation $a_{n} = 2a_{n-1} + 3 \cdot 2^{n}.$
9
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.]
10
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 $a_{1} = 56\:\text{and}\: a_{2} = 278.$
11
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 $a_{1} = 5.$
12
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 $a_{1} = 4.$
13
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 $F(n) = n^{3}?$ $F(n) = (-2)^{n}?$ $F(n) = n2^{n}?$ $F(n) = n^{2}4^{n}?$ $F(n) = (n^{2} - 2)(-2)^{n}?$ $F(n) = n^{4}2^{n}?$ $F(n) = 2?$
14
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 $F (n) = n^{2}?$ $F (n) = 2^{n}?$ $F (n) = n2^{n}?$ $F (n) = (-2)^{n}?$ $F (n) = n^{2}2^{n}?$ $F (n) = n^{3}(-2)^{n}?$ $F (n) = 3?$
15
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 $a_{0} = 4.$
16
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 $a_{0} = 2.$
17
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 $a_{0} = 1.$
18
What is the general form of the solutions of a linear homogeneous recurrence relation if its characteristic equation has the roots $-1, -1, -1, 2, 2, 5, 5, 7?$
19
What is the general form of the solutions of a linear homogeneous recurrence relation if its characteristic equation has roots $1,1,1,1,−2,−2,−2,3,3,−4?$
20
Find the general form of the solutions of the recurrence relation $a_{n} = 8a_{n−2} − 16a_{n−4}.$
21
Solve the recurrence relation $a_{n} = −3a_{n−1} − 3a_{n−2} − a_{n−3}\:\text{with}\: a_{0} = 5, a_{1} = −9,\:\text{and}\: a_{2} = 15.$
22
Solve the recurrence relation $a_{n} = 6a_{n−1} − 12a_{n−2} + 8a_{n−3} \:\text{with}\: a_{0} = −5, a_{1} = 4,\: \text{and}\: a_{2} = 88.$
23
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.]
24
Prove Theorem $3:$ Let $c_{1},c_{2},\dots,c_{k}$ be real numbers. Suppose that the characteristic equation $r^{k}-c_{1}r^{k-1}-\dots - c_{k} = 0$ 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.
25
Find the solution to $a_{n} = 2a_{n−1} + 5a_{n−2} − 6a_{n−3}\: \text{with}\: a_{0} = 7, a_{1} = −4,\:\text{and}\: a_{2} = 8.$
26
Find the solution to $a_{n} = 5a_{n−2}− 4a_{n−4} \:\text{with}\: a_{0} = 3, a_{1} = 2, a_{2} = 6, \:\text{and}\: a_{3} = 8.$
27
Find the solution to $a_{n} = 7a_{n−2} + 6a_{n−3}\:\text{with}\: a_{0} = 9, a_{1} = 10, \text{and}\: a_{2} = 32.$
Find the solution to $a_{n} = 2a_{n−1} + a_{n−2} − 2a_{n−3} \:\text{for}\: n = 3, 4, 5,\dots, \:\text{with}\: a_{0} = 3, a_{1} = 6, \:\text{and}\: a_{2} = 0.$
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.
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.