Answer (d) only. Problem 1Suppose A = (an) = (a1, a2, az,-...) is an increasing sequence of positive integers.A number e is called A-erpressible if c is the alternating sum of a finite subsequenceof A. To form such a sum, choose a finite subset of the sequence A, list those umbersin increasing order (no repetitions allowed), and combine them with alternating plusand minus signs. We allow the trivial case of one-element subsequences, so that eacha, is A-expressible.Definition. Sequence A = (a,n) is an "alt-basis" if every positive integer is uniquelyA-expressible. That is, for every integer m > 0, there is exactly one way to expressm as an alternating sum of a finite subsequence of A.Examples. Sequence B = (2"-') = (1, 2, 4, 8, 16,...) is not an alt-basis becausesome numbers are B-expressible in more than one way. For instance 3 = -1+4 =1-2+4.Sequence C = (3"-1) = (1, 3, 9, 27, 81,...) is not an alt-basis because some numbers(like 4 and 5) are not C-expressible.(a) Let D = (2" – 1) = (1, 3, 7, 15, 31, ...). Note that:1 = 1, 2= -1+3, 3 = 3, 4= -3+7, 5=1-3+7,6 = -1+7, 7 = 7, 8 = -7+ 15, 9 = 1 – 7+ 15, ...Prove that D is an alt-basis.(b) Can some E = (4, 5, 7,...) be an alt-basis? That is, does there exist analt-basis E = (en) with e = 4, ez = 5, and ez = 7?Justify your answer.The first few values seem to work: 1 = -4 + 5, 2=-5+ 7, 3= -4+ 7.(c) Can F = (1, 4,...) be an alt-basis? That is, does there exist an alt-basis F = (fn)with fi = 1 and f2 = 4?(d) Investigate some other examples. Is there some fairly simple test to determinewhether a given sequence A = (an) is an alt-basis?

Name: Answer (d) only. Problem 1Suppose A = (an) = (a1, a2, az,-...) is an i
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Description: Answer (d) only. Problem 1Suppose A = (an) = (a1, a2, az,-...) is an increasing sequence of positive integers.A number e is called A-erpressible if c is the alternating sum of a finite subsequenceof A. To form such a sum, choose a finite subset of th

A = an = 2n = ( 1,2,4,8,16 . . . ) is an increasing sequence of positive integer. 1 = 1 2 = 2 3 =…

Suppose A = (a,) = (a1, a2,as....) is an increasing sequence of positive integers. A number e is called A-crpressible if e is the alternating sum of a finite subsequence of A. To form such a sum, choose a finite subset of the sequence A, list those mumbers in increasing order (no repetitions allowed), and combine them with alternating plus and minus signs. We allow the trivial case of one-element subsequences, so that each a, is A-expressible. Definition. Sequence A = (a.) is an "alt-basis" if every positive integer is uniquely A-expressible. That is, for every integer m > 0, there is exactly one way to express m as an alternating sum of a finite subsequence of A. (2-) (1, 2, 4, 8, 16,...) is not an alt-basis because Examples. Sequence B = some numbers are B-expressible in more than one way. For instance 3 -1+4 = 1-2+4. Sequence C = (3-) = (1, 3, 9, 27, 81,...) is not an alt-basis because some numbers (like 4 and 5) are not C-expressible. (a) Let D = (2" – 1) = (1, 3, 7, 15, 31,...). Note that: 1= 1, 2= -1 +3, 3= 3, 4 =-3+7, 5=1-3+7, 6 = -1+7, 7 = 7, 8 = -7+ 15, 9 =1-7+ 15, Prove that D is an alt-basis. (b) Can some E = (4, 5,7,...) be an alt-basis? That is, does there exist an alt-basis E = (e,) with e = 4, ez = 5, and e = 7? Justify your answer. The first few values seem to work: 1=-4+5, 2= -5 +7, 3= -4+7. (c) Can F = (1, 4,...) be an alt-basis? That is, does there exist an alt-basis F = (fm) with fi =1 and fa = 4? (d) Investigate some other examples. Is there some fairly simple test to determine whether a given sequence A = (an) is an alt-basis?

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Answer (d) only.

Problem 1
Suppose A = (an) = (a1, a2, az,-...) is an increasing sequence of positive integers.
A number e is called A-erpressible if c is the alternating sum of a finite subsequence
of A. To form such a sum, choose a finite subset of the sequence A, list those umbers
in increasing order (no repetitions allowed), and combine them with alternating plus
and minus signs. We allow the trivial case of one-element subsequences, so that each
a, is A-expressible.
Definition. Sequence A = (a,n) is an "alt-basis" if every positive integer is uniquely
A-expressible. That is, for every integer m > 0, there is exactly one way to express
m as an alternating sum of a finite subsequence of A.
Examples. Sequence B = (2"-') = (1, 2, 4, 8, 16,...) is not an alt-basis because
some numbers are B-expressible in more than one way. For instance 3 = -1+4 =
1-2+4.
Sequence C = (3"-1) = (1, 3, 9, 27, 81,...) is not an alt-basis because some numbers
(like 4 and 5) are not C-expressible.
(a) Let D = (2" – 1) = (1, 3, 7, 15, 31, ...). Note that:
1 = 1, 2= -1+3, 3 = 3, 4= -3+7, 5=1-3+7,
6 = -1+7, 7 = 7, 8 = -7+ 15, 9 = 1 – 7+ 15, ...
Prove that D is an alt-basis.
(b) Can some E = (4, 5, 7,...) be an alt-basis? That is, does there exist an
alt-basis E = (en) with e = 4, ez = 5, and ez = 7?
Justify your answer.
The first few values seem to work: 1 = -4 + 5, 2=-5+ 7, 3= -4+ 7.
(c) Can F = (1, 4,...) be an alt-basis? That is, does there exist an alt-basis F = (fn)
with fi = 1 and f2 = 4?
(d) Investigate some other examples. Is there some fairly simple test to determine
whether a given sequence A = (an) is an alt-basis?

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