本次美国代写主要为离散数学相关的assignment

1.(2 分)给出下列数列的第 9 项和第 10 项。
(a) fang1 n=1 其中 an = ( 1)n + ( 2)n 1
(b) fbng1 n=1 其中 bn = 12 + 3n + n2

2.(4 分)对于以下每个序列,提供一个描述它们的简单规则。找到规则后,请给出序列中的下两个元素。注意,序列从 n = 1 到 nity。
(a) 1; 1; 2; 2; 2; 3; 3; 4; 4; 4; 5; 5; 6; 6; 6; : : :
(b) 0; 3; 8; 15; 24; 35; 48; 63::::

3.(3 分)对于以下每个序列,请指定以下哪些属性适用于它们:递增、非递减、严格递增、递减、非递增和/或严格递减。
(a) 9; 7; 5; 3; 1; 1; : : :
(b) 1; 1; 2; 2; 2; 3; 3; 4; 4; 4; 5; 5; 6; 6; 6; : : :
(c) 1; 1; 1; 1; 1; 1; : : :

4.(6 分)对于以下每个序列,确定它们是算术的、几何的,还是两者都不是。对于算术和/或几何的那些,提供共同的差异和/或比率。
(a) 9; 7; 5; 3; 1; 1; : : :
(b) 2; 10; 50; 250; : : :
(c) 729; 243; 81; 27; 9; : : :

5.(6 分)对于下面的每一组,判断它是 nite,可数 in nite,还是
不可数。对于可数在有限范围内的那些,表现出正集合之间的双射
整数和那个集合。
(a) 完全平方数的整数集。
(b) 作为整数平方根的实数集。
(c) 0 到 1 之间的实数集。

6.(9分)考虑猜想:
n+1 表示 n 1 其中 n 2 Z+。这个猜想可以
通过归纳法证明。
(a) 陈述归纳证明的基本情况。
(b) 证明基本情况是正确的。
(c) 陈述归纳假设。
(d) 陈述我们必须在归纳步骤中证明的猜想。
(e) 完成归纳步骤的证明。

1. (2 points) Give the 9th and 10th terms of the following sequences.
(a) fang1 n=1 where an = ( 1)n + ( 2)n 1
(b) fbng1 n=1 where bn = 12 + 3n + n2

2. (4 points) For each of the following sequences, provide a simple rule that describes them. Once you have found the rule, give the next two elements in the sequence. Note, the sequences go from n = 1 to in nity.
(a) 1; 1; 2; 2; 2; 3; 3; 4; 4; 4; 5; 5; 6; 6; 6; : : :
(b) 0; 3; 8; 15; 24; 35; 48; 63 : : :

3. (3 points) For each of the following sequences, specify which of the following properties apply to them: increasing, non-decreasing, strictly increasing, decreasing, non-increasing, and/or strictly decreasing.
(a) 9; 7; 5; 3; 1; 1; : : :
(b) 1; 1; 2; 2; 2; 3; 3; 4; 4; 4; 5; 5; 6; 6; 6; : : :
(c) 1; 1; 1; 1; 1; 1; : : :

4. (6 points) For each of the following sequences, determine if they are arithmetic, geometric, both or neither. For those that are arithmetic and/or geometric, provide the common di erence and/or ratio.
(a) 9; 7; 5; 3; 1; 1; : : :
(b) 2; 10; 50; 250; : : :
(c) 729; 243; 81; 27; 9; : : :

5. (6 points) For each of the following sets, determine whether it is nite, countably in nite, or
uncountable. For those that are countably in nite, exhibit a bijection between the set of positive
integers and that set.
(a) The set of integers that are perfect squares.
(b) The set of real numbers that are the square roots of integers.
(c) The set of real numbers between 0 and 1.

6. (9 points) Consider the conjecture:
n+1 for n  1 where n 2 Z+. This conjecture can
be proven via induction.
(a) State the base case for a proof by induction.
(b) Show that the base case is true.
(c) State the inductive hypothesis.
(d) State the conjecture that we must prove in the inductive step.
(e) Complete the proof of the inductive step.


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