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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?

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
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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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?
Transcribed Image Text: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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