Suppose f= (J fa Ss.) is a sequence of integers. For 0 5kSn, define the "J-bin" mumbers [2), as follows: Define (3), = 1, and for k 21 let *** JA If , =n, then (2), - () is the usual binomial coeficient. Another example: Define I (1,3, 4,7,11, 18,..) by setting La = 1, La = 3 and - L+ Lana forn> 2. Here are lists of some of the f-bin mumbers [(:), for the sequences f = I and f= L. 2 3 2 21 S S 10 10 SI IS 20 61 1-hin er L-hin mumber Definition. Sequence fis binomid if all the f-bin numbers [), are integers. Equivalently: / is binomid when, for cach k 2 1: Every product of k consecutive terms fafa-i fa-t+i is an integer multiple of the product of the first k consecutive terms f.fa-i Since every binomial coeficient () is an integer, the sequence / is binomid. The table above shows that the sequence L is not binomid. (a) Define sequences P, = 2" = (2, 4, 8,..). Q, = n = (1, 4,9.), and D, = 2n = (2, 4, 6,..). In each case, find a simple formula for (2), check that it is an integer, and conclude that P, Q and D are binomid.
Suppose f= (J fa Ss.) is a sequence of integers. For 0 5kSn, define the "J-bin" mumbers [2), as follows: Define (3), = 1, and for k 21 let *** JA If , =n, then (2), - () is the usual binomial coeficient. Another example: Define I (1,3, 4,7,11, 18,..) by setting La = 1, La = 3 and - L+ Lana forn> 2. Here are lists of some of the f-bin mumbers [(:), for the sequences f = I and f= L. 2 3 2 21 S S 10 10 SI IS 20 61 1-hin er L-hin mumber Definition. Sequence fis binomid if all the f-bin numbers [), are integers. Equivalently: / is binomid when, for cach k 2 1: Every product of k consecutive terms fafa-i fa-t+i is an integer multiple of the product of the first k consecutive terms f.fa-i Since every binomial coeficient () is an integer, the sequence / is binomid. The table above shows that the sequence L is not binomid. (a) Define sequences P, = 2" = (2, 4, 8,..). Q, = n = (1, 4,9.), and D, = 2n = (2, 4, 6,..). In each case, find a simple formula for (2), check that it is an integer, and conclude that P, Q and D are binomid.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Answer (a), (b), and (c) only.
![Suppose f = (f1. fa fs...) is a sequence of integers. For 0<k<n, define the
"J-bin" numbers (2), as follows: Define [8), = 1, and for k 2 1 let
If I, =n, then (C), = (:) is the usual binomial coefficient.
Another example: Define L = (1,3, 4, 7, 11, 18,...) by setting La = 1, L2 = 3 and
L, = L- + La-2 for n> 2.
Here are lists of some of the f-bin numbers [E), for the sequences f = I and f = L.
01 2 3 4 56
|0 12 3
345 6
21 2 I
31 33
1
5I5 10 10 SI
16 15 20 15 6 1
: :: : :::
sli 11 .
6 18 66
::: :
1-bin sumber
L-hin number
Definition. Sequence f is binomid if all the f-bin numbers [2], are integers.
Equivalently: f is binomid when, for each k > 1:
Every product of k consecutive terms fafn-l. fn-k+1 is an integer multiple of the
product of the first k consecutive terms fafk-1 fi-
Since every binomial coefficient (C) 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, ...), Qa = n? = (1,4,9,...), and
D. = 2n = (2, 4, 6,...). In each case, find a simple formula for [7), 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?
As a first step, verify that [), = = ". is always an integer.
(d) Find some further examples of binomid sequences. Are there some interesting
conditions on a sequence f that imply that f is binomid?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8dc465b5-f02f-4e4a-a6bc-8e6eeff88fe8%2Fc5e21641-7e47-4be0-95b9-edf371260261%2Fvxuv9c4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose f = (f1. fa fs...) is a sequence of integers. For 0<k<n, define the
"J-bin" numbers (2), as follows: Define [8), = 1, and for k 2 1 let
If I, =n, then (C), = (:) is the usual binomial coefficient.
Another example: Define L = (1,3, 4, 7, 11, 18,...) by setting La = 1, L2 = 3 and
L, = L- + La-2 for n> 2.
Here are lists of some of the f-bin numbers [E), for the sequences f = I and f = L.
01 2 3 4 56
|0 12 3
345 6
21 2 I
31 33
1
5I5 10 10 SI
16 15 20 15 6 1
: :: : :::
sli 11 .
6 18 66
::: :
1-bin sumber
L-hin number
Definition. Sequence f is binomid if all the f-bin numbers [2], are integers.
Equivalently: f is binomid when, for each k > 1:
Every product of k consecutive terms fafn-l. fn-k+1 is an integer multiple of the
product of the first k consecutive terms fafk-1 fi-
Since every binomial coefficient (C) 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, ...), Qa = n? = (1,4,9,...), and
D. = 2n = (2, 4, 6,...). In each case, find a simple formula for [7), 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?
As a first step, verify that [), = = ". is always an integer.
(d) Find some further examples of binomid sequences. Are there some interesting
conditions on a sequence f that imply that f is binomid?
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