Answer (d) only. Problem 1Suppose A = (an) = (a1, a2, az3, ...) is an increasing sequence of positive integers.A number c 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 numbersin increasing order (no repetitions allowed), and combine them with alternating plusand minus signs. We allow the trivial case of one-element subsequences, so that eachan is A-expressible.Definition. Sequence A = (an) 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) = (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 e1 = 4, e2 = 5, and e3 = 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, az3, ...) is an
Uploaded: 2024-03-20T06:47:08.000Z
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Description: Answer (d) only. Problem 1Suppose A = (an) = (a1, a2, az3, ...) is an increasing sequence of positive integers.A number c 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 t