Problem 4 Suppose f (f1, f2, f3, -..) is a sequence of integers. For 0 2. Here are lists of some of the f-bin numbers [], for the sequences f = I and f = L. 1 2 3 4 5 6 3 4 1 1 1 1 1 2 1 2 1 1 1. 3 1 3 3 1 1 4 1. 4 1 4 6 1 1 28 7 1 1 5 10 10 1 1 11 11 1. 1 6 學 15 20 15 6 1. 6. 18 66 66 18 1 1- bin numbers L- bin numbers Definition. Sequence f is binomid if all the f-bin numbers [), are integers. Equivalently: f is binomid when, for each k 2 1: Every product of k consecutive terms fnfn-1.. fn-k+1 is an integer multiple of the product of the first k consecutive terms frfk-1. f1. Since every binomial coefficient (") is an integer, the sequence I is binomid. The table above shows that the sequence L is not binomid. (a) Define sequences P = 2" = (2,4,8, . ), Qn = n² = (1,4, 9,...), and D, = 2n = (2, 4, 6, ...). In each case, find a simple formula for [), check that it is an integer, and conclude that P, Q and D are binomid. (b) Is the sequence M, = 2" -1 binomid? Justify your answer. (c) Is the sequence T, = n(n +1) binomid? T, Tn-1 As a first step, verify that ["], = n(n+1) , (n-1)n is always an integer. %3D 6 (d) Find some further examples of binomid sequences. Are there some interesting conditions on a sequence f that imply thatf is binomid?
Problem 4 Suppose f (f1, f2, f3, -..) is a sequence of integers. For 0 2. Here are lists of some of the f-bin numbers [], for the sequences f = I and f = L. 1 2 3 4 5 6 3 4 1 1 1 1 1 2 1 2 1 1 1. 3 1 3 3 1 1 4 1. 4 1 4 6 1 1 28 7 1 1 5 10 10 1 1 11 11 1. 1 6 學 15 20 15 6 1. 6. 18 66 66 18 1 1- bin numbers L- bin numbers Definition. Sequence f is binomid if all the f-bin numbers [), are integers. Equivalently: f is binomid when, for each k 2 1: Every product of k consecutive terms fnfn-1.. fn-k+1 is an integer multiple of the product of the first k consecutive terms frfk-1. f1. Since every binomial coefficient (") is an integer, the sequence I is binomid. The table above shows that the sequence L is not binomid. (a) Define sequences P = 2" = (2,4,8, . ), Qn = n² = (1,4, 9,...), and D, = 2n = (2, 4, 6, ...). In each case, find a simple formula for [), check that it is an integer, and conclude that P, Q and D are binomid. (b) Is the sequence M, = 2" -1 binomid? Justify your answer. (c) Is the sequence T, = n(n +1) binomid? T, Tn-1 As a first step, verify that ["], = n(n+1) , (n-1)n is always an integer. %3D 6 (d) Find some further examples of binomid sequences. Are there some interesting conditions on a sequence f that imply thatf is binomid?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Topic Video
Question
100%
Answer (d) only.
![Problem 4
Suppose f = (fi, f2, f3,...) is a sequence of integers. For 0 < k < n, define the
"f-bin" numbers [%],
as follows: Define [8],= 1, and for k >1 let
[:),
fn fn-1·*· fn-k+1
fk fk-1
fi
...
(") is the usual binomial coefficient.*
If I, = n, then [%),
Another example: Define L = (1,3, 4, 7, 11, 18, ...) by setting Lị = 1, L2 = 3 and
Ln = Ln-1+ Ln-2 for n > 2.
Here are lists of some of the f-bin numbers [), for the sequences f = I and f = L.
1
2
4.
5 6
1
2
3
5
1
1
1
1
1
1
1
1
2
1
1.
3
1
3
3.
1
3
1
4
4
4
1.
4
6
4
4
1
7
1
5
1
10
10
1
5
11
11
1.
1
15
20
15
1.
1
18
66
231
66
18
1.
I- bin numbers
L- bin numbers
Definition. Sequence f is binomid if all the f-bin numbers [),
are integers.
Equivalently: f is binomid when, for each k > 1:
Every product of k consecutive terms fnfn-1· fn-k+1 is an integer multiple of the
product of the first k consecutive terms fkfk-1… f1-
Since every binomial coefficient () is an integer, the sequence I is binomid. The
table above shows that the sequence L is not binomid.
(a) Define sequences P = 2" = (2,4, 8, . ..), Qn = n² = (1, 4, 9, . ..), and
Dn = 2n = (2, 4, 6, . . ). In each case, find a simple formula for [), check that it
is an integer, and conclude that P, Q and D are binomid.
(b)
the sequence Mn = 2" – 1 binomid? Justify your answer.
(c) Is the sequence T,
= n(n + 1) binomid?
As a first step, verify that [" = T, Tn-1
n(n+1) , (n-1)n is always an integer.
6
T2 T1
2
(d) Find some further examples of binomid sequences. Are there some interesting
conditions on a sequence f that imply that f is binomid?
*Other notations for (") include „C and C(n, k).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8dc465b5-f02f-4e4a-a6bc-8e6eeff88fe8%2Fff15fb05-66ae-43de-85e9-76b639dcc25c%2F70e5ptnh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 4
Suppose f = (fi, f2, f3,...) is a sequence of integers. For 0 < k < n, define the
"f-bin" numbers [%],
as follows: Define [8],= 1, and for k >1 let
[:),
fn fn-1·*· fn-k+1
fk fk-1
fi
...
(") is the usual binomial coefficient.*
If I, = n, then [%),
Another example: Define L = (1,3, 4, 7, 11, 18, ...) by setting Lị = 1, L2 = 3 and
Ln = Ln-1+ Ln-2 for n > 2.
Here are lists of some of the f-bin numbers [), for the sequences f = I and f = L.
1
2
4.
5 6
1
2
3
5
1
1
1
1
1
1
1
1
2
1
1.
3
1
3
3.
1
3
1
4
4
4
1.
4
6
4
4
1
7
1
5
1
10
10
1
5
11
11
1.
1
15
20
15
1.
1
18
66
231
66
18
1.
I- bin numbers
L- bin numbers
Definition. Sequence f is binomid if all the f-bin numbers [),
are integers.
Equivalently: f is binomid when, for each k > 1:
Every product of k consecutive terms fnfn-1· fn-k+1 is an integer multiple of the
product of the first k consecutive terms fkfk-1… f1-
Since every binomial coefficient () is an integer, the sequence I is binomid. The
table above shows that the sequence L is not binomid.
(a) Define sequences P = 2" = (2,4, 8, . ..), Qn = n² = (1, 4, 9, . ..), and
Dn = 2n = (2, 4, 6, . . ). In each case, find a simple formula for [), check that it
is an integer, and conclude that P, Q and D are binomid.
(b)
the sequence Mn = 2" – 1 binomid? Justify your answer.
(c) Is the sequence T,
= n(n + 1) binomid?
As a first step, verify that [" = T, Tn-1
n(n+1) , (n-1)n is always an integer.
6
T2 T1
2
(d) Find some further examples of binomid sequences. Are there some interesting
conditions on a sequence f that imply that f is binomid?
*Other notations for (") include „C and C(n, k).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
