Login
Register
@
Dark Mode
Profile
Edit my Profile
Messages
My favorites
Register
Activity
Q&A
Questions
Unanswered
Tags
Subjects
Users
Ask
Previous Years
Blogs
New Blog
Exams
Dark Mode
Recent questions tagged discrete-mathematics
0
votes
0
answers
121
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.
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
146
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
122
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.
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
105
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
1
answer
123
Kenneth Rosen Edition 7 Exercise 8.2 Question 35 (Page No. 526)
Find the solution of the recurrence relation $a_{n} = 4a_{n-1} - 3a_{n-2} + 2^{n} + n + 3\:\text{with}\: a_{0} = 1\:\text{and}\: a_{1} = 4.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
212
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
124
Kenneth Rosen Edition 7 Exercise 8.2 Question 34 (Page No. 526)
Find all solutions of the recurrence relation $a_{n} =7a_{n-1} - 16a_{n-2} + 12a_{n-3} + n4^{n}\:\text{with}\: a_{0} = -2,a_{1} = 0,\:\text{and}\: a_{2} = 5.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
123
views
kenneth-rosen
discrete-mathematics
descriptive
counting
recurrence-relation
0
votes
1
answer
125
Kenneth Rosen Edition 7 Exercise 8.2 Question 33 (Page No. 525)
Find all solutions of the recurrence relation $a_{n} = 4a_{n-1} - 4a_{n-2} + (n + 1)2^{n}.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
277
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
126
Kenneth Rosen Edition 7 Exercise 8.2 Question 32 (Page No. 525)
Find the solution of the recurrence relation $a_{n} = 2a_{n-1} + 3 \cdot 2^{n}.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
139
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
127
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.]
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
119
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
128
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 $a_{1} = 56\:\text{and}\: a_{2} = 278.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
201
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
129
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 $a_{1} = 5.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
136
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
130
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 $a_{1} = 4.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
112
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
131
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}? $ $F(n) = n^{2}4^{n}?$ $F(n) = (n^{2} - 2)(-2)^{n}?$ $F(n) = n^{4}2^{n}?$ $F(n) = 2?$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
102
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
132
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 $F (n) = n^{2}?$ $F (n) = 2^{n}?$ $F (n) = n2^{n}?$ $F (n) = (-2)^{n}?$ $F (n) = n^{2}2^{n}?$ $F (n) = n^{3}(-2)^{n}?$ $F (n) = 3?$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
125
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
133
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 $a_{0} = 4.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
113
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
134
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 $a_{0} = 2.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
109
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
135
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 $a_{0} = 1.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
138
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
1
answer
136
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 $-1, -1, -1, 2, 2, 5, 5, 7?$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
1.2k
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
1
answer
137
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 $1,1,1,1,−2,−2,−2,3,3,−4?$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
156
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
1
answer
138
Kenneth Rosen Edition 7 Exercise 8.2 Question 20 (Page No. 525)
Find the general form of the solutions of the recurrence relation $a_{n} = 8a_{n−2} − 16a_{n−4}.$
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
169
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
1
answer
139
Kenneth Rosen Edition 7 Exercise 8.2 Question 19 (Page No. 525)
Solve the recurrence relation $a_{n} = −3a_{n−1} − 3a_{n−2} − a_{n−3}\:\text{with}\: a_{0} = 5, a_{1} = −9,\:\text{and}\: a_{2} = 15.$
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
187
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
1
answer
140
Kenneth Rosen Edition 7 Exercise 8.2 Question 18 (Page No. 525)
Solve the recurrence relation $a_{n} = 6a_{n−1} − 12a_{n−2} + 8a_{n−3} \:\text{with}\: a_{0} = −5, a_{1} = 4,\: \text{and}\: a_{2} = 88.$
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
128
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
141
Kenneth Rosen Edition 7 Exercise 8.2 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 $k = n/2 .$ ... Show that the sequence $\{a_{n}\}$ satisfies the same recurrence relation and initial conditions satisfied by the sequence of Fibonacci numbers.]
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
106
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
142
Kenneth Rosen Edition 7 Exercise 8.2 Question 16 (Page No. 525)
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.
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
115
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
1
answer
143
Kenneth Rosen Edition 7 Exercise 8.2 Question 15 (Page No. 525)
Find the solution to $a_{n} = 2a_{n−1} + 5a_{n−2} − 6a_{n−3}\: \text{with}\: a_{0} = 7, a_{1} = −4,\:\text{and}\: a_{2} = 8.$
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
123
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
1
answer
144
Kenneth Rosen Edition 7 Exercise 8.2 Question 14 (Page No. 525)
Find the solution to $a_{n} = 5a_{n−2}− 4a_{n−4} \:\text{with}\: a_{0} = 3, a_{1} = 2, a_{2} = 6, \:\text{and}\: a_{3} = 8.$
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
112
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
1
answer
145
Kenneth Rosen Edition 7 Exercise 8.2 Question 13 (Page No. 525)
Find the solution to $a_{n} = 7a_{n−2} + 6a_{n−3}\:\text{with}\: a_{0} = 9, a_{1} = 10, \text{and}\: a_{2} = 32.$
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
107
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
1
answer
146
Kenneth Rosen Edition 7 Exercise 8.2 Question 12 (Page No. 525)
Find the solution to $a_{n} = 2a_{n−1} + a_{n−2} − 2a_{n−3} \:\text{for}\: n = 3, 4, 5,\dots, \:\text{with}\: a_{0} = 3, a_{1} = 6, \:\text{and}\: a_{2} = 0.$
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
125
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
147
Kenneth Rosen Edition 7 Exercise 8.2 Question 11 (Page No. 525)
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.
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
111
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
148
Kenneth Rosen Edition 7 Exercise 8.2 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 $\{a_{n}\}$ ... $n = 0,1,2,\dots,$ where $\alpha_{1}$ and $\alpha_{2}$ are constants.
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
126
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
proof
0
votes
0
answers
149
Kenneth Rosen Edition 7 Exercise 8.2 Question 9 (Page No. 525)
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 $ ... $n$ years if no money has been withdrawn?
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
313
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
0
votes
0
answers
150
Kenneth Rosen Edition 7 Exercise 8.2 Question 8 (Page No. 524 - 525)
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. Find a recurrence relation for $\{L_{n}\},$ ... if $100,000$ lobsters were caught in year $1\:\text{ and}\: 300,000$ were caught in year $2.$
Lakshman Patel RJIT
asked
in
Combinatory
May 3, 2020
by
Lakshman Patel RJIT
154
views
kenneth-rosen
discrete-mathematics
counting
recurrence-relation
descriptive
Page:
« prev
1
2
3
4
5
6
7
8
9
10
...
75
next »
Subscribe to GATE CSE 2023 Test Series
Subscribe to GO Classes for GATE CSE 2023
Quick search syntax
tags
tag:apple
author
user:martin
title
title:apple
content
content:apple
exclude
-tag:apple
force match
+apple
views
views:100
score
score:10
answers
answers:2
is accepted
isaccepted:true
is closed
isclosed:true
Recent Posts
ISRO RECRUITMENT FOR SCIENTIST B THROUGH GATE
POWER GRID CORPORATION OF INDIA LIMITED
INSTITUTE OF BANKING PERSONNEL SELECTION
GATE Overflow books for TIFR, ISRO, UGCNET and NIELIT
RECRUITMENT IN OIL AND GAS CORPORATION LIMITED
Subjects
All categories
General Aptitude
(2.4k)
Engineering Mathematics
(9.1k)
Digital Logic
(3.2k)
Programming and DS
(5.8k)
Algorithms
(4.5k)
Theory of Computation
(6.6k)
Compiler Design
(2.3k)
Operating System
(4.9k)
Databases
(4.5k)
CO and Architecture
(3.7k)
Computer Networks
(4.5k)
Non GATE
(1.3k)
Others
(2.4k)
Admissions
(647)
Exam Queries
(841)
Tier 1 Placement Questions
(17)
Job Queries
(74)
Projects
(9)
Unknown Category
(855)
Recent questions tagged discrete-mathematics
Recent Blog Comments
@abir_banerjee Thanks Abir. I'm third year...
@nolan_keats Currently I am in third year...
@abir_banerjee thank you Abir.Supposing you...
@nolan_keats just a suggestion as I also...
@abir_banerjee Hope I can do this in span of one...