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RSS订阅原创 Introduction to Mathematical Thinking-Problem 10
10.Give an example of a family of intervals AnA_nAn, n=1,2,...n=1,2,...n=1,2,..., such that An+1⊂AnA_{n+1}\subset A_nAn+1⊂An for all n and ∪n=1∞An\displaystyle \cup_{n=1}^\infty A_n∪n=1∞An consis...
2020-02-02 15:00:21
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原创 Introduction to Mathematical Thinking-Problem 9
9.Give an example of a family of intervals AnA_nAn such that An+1⊂AnA_{n+1}\subset A_nAn+1⊂An for all nnn and ⋂n=1∞An=∅\displaystyle \bigcap_{n=1}^\infty A_n=\emptysetn=1⋂∞An=∅.Proof: Set An=(0,...
2020-02-02 14:41:36
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原创 Introduction to Mathematical Thinking-Problem 8
8.Prove that if the sequence {an}n=1∞\left\{a_n\right\}_{n=1}^\infty{an}n=1∞ tends to limit L as n→∞n \rightarrow \inftyn→∞, then for any fixed number M>0M>0M>0, the sequence {Man}n=1∞\left...
2020-02-02 14:28:42
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原创 Introduction to Mathematical Thinking-Problem 7
7.Prove that for any natural number nnn, 2+22+23......+2n=2n+1−22+2^2+2^3......+2^n=2^{n+1}-22+22+23......+2n=2n+1−2.Proof: Let the left hand side of the identity be SSS. We have 2S=22+23......+2n+2n...
2020-02-02 14:25:55
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原创 Introduction to Mathematical Thinking-Problem 6
6.Prove that the only prime triple is 3,5,7.Proof: As we have proved in problem 5, there must be a multiple of 3 in the triple n,n+2,n+4n, n+2, n+4n,n+2,n+4. Thus the element must be 3, or else it is...
2020-02-02 13:42:58
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原创 Introduction to Mathematical Thinking-Problem 5
5.Prove that for any integer nnn, at least one of the integers n,n+2,n+4n, n+2, n+4n,n+2,n+4 is divisible by 3.Proof: By the Division Theorem, nnn can be expressed as either 3q3q3q, or 3q+13q+13q+1, ...
2020-02-02 13:31:37
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原创 Introduction to Mathematical Thinking-Problem 4
4.Prove that every odd natural number is of one of the forms 4n+14n+14n+1 or 4n+34n+34n+3, where nnn is an integer.Proof: By the Division Theorem, we can express any natural number in the form of 4q+...
2020-02-02 13:28:53
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原创 Introduction to Mathematical Thinking-Problem 3
3.Say whether the following is true or false and support your answer by a proof.For any integer nnn, the number n2+n+1n^2+n+1n2+n+1 is odd.Proof: n2+n+1=n(n+1)+1n^2+n+1=n(n+1)+1n2+n+1=n(n+1)+1. As n...
2020-02-02 13:27:43
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原创 Introduction to Mathematical Thinking-Problem 2
2.Say whether the following is true or false and support your answer by a proof.The sum of any five consecutive integers is divisible by 5.Proof: Let nnn be any integer, we shall prove that the sum ...
2020-02-02 13:25:11
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原创 Introduction to Mathematical Thinking-Problem 1
Introduction to Mathematical ThinkingProblem 1: Say whether the following is true or false and support your answer by a proof.(∃m∈N)(∃n∈N)(3m+5n=12)(\exists m \in \mathbb{N})(\exists n \in \mathbb{N...
2020-02-02 13:22:57
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