利用等差数列的性质解题几例

等差数列有不少性质,现列举如下: 一、设{a,},{b。}是等差数列,则{a。+b。}也是等差数列. 二、设{a,}是等差数列,。为常数,则{ca,}也是等差数列. 三、设{a。}是等差数列,c为常数,则{a。+好也是等差数列. 四、设{a。}是等差数列,如果项数号码P,q,:,:,…成等差数列,则。,,a。,。r,a,,…也成等差数列. 以上性质皆不难证明,仅就性质四为例来证明。 设等差数列通a,}的公差为d,取新数列a,,a。,a,一中的任意相邻两项“‘,a,, az一a、=〔a;+(j一1)d)一〔a:+(‘一l)d〕 =(j一f)d. ,.’p,q,:…i,j,…成等差数列,+abc也成等差数差列,各项都减去abc,则 …     

也谈数列一个性质的证明

<正>如果各项均为正数的等比数列和一个等差数列,首项、末项、项数分别相等,那么等比数列各项之和不超过等差数列各项之和.这是文[1]给出的关于数列的一个性质,其证明较为复杂.经研究,笔者得到一个更强的结论:如果各项均为正数的等比数列和一个等差数列,首项、末项、项数分别相等,那么等比数列的各项均不超过等差数列对应的项.     

数学课是这样“备”出来的

2006年4月4日,笔者参加了上海市青年教师教学优质课评比,获得一等奖.上课内容为“等差数列的前n项和公式(一)”.本文围绕着这节课的设计、试教及修改的全过程,谈谈本人在二期课改背景下对课堂教学设计的一点体会.等差数列是高中数学研究的两个基本数列之一.等差数列的前n项和公式则是等差数列中的一个重要公式.它前承等差数列的定义、通项公式,后启等比数列的前n项和公式.本节课是数列求和的第一课,同时也是“倒序相加法”这一重要求和方法的典型载体.本课的教学重点是两个:(1)探究并获得等差数列的前n项和公式;(2)等差数列前n项和公式的初…     

例谈等差与等比数列的类比推理

<正>我通过研究近几年的高考试题,发现类比推理的考查较为突出,是高考的一个新的亮点.但是如何将知识进行类比,对于学生来说是一个难点.通过研究相关题型,我总结出等差数列和等比数列类比的规律:等差数列的公差对应等比数列的公比,等差数列的加减法运算对应等比数列的乘除法运算,等差数列的乘除法运算对应等比数列的乘方开方运算.本文将结合一些例子谈谈如何应用该规律对等差数列和等比数列间进行类比.     

数列

1.本单元重、难点分析本单元的重点:等差数列、等比数列的概念、通项公式及前n项和公式,等差数列、等比数列的有关性质及其应用.本单元的难点:等差数列、等比数列的通项公式与前n项和公式的推导以及它们的综合运用.在数列的五个基本量(等差数列中:a1,n,d,an,Sn;等比数列中:a1,q,n,an,Sn)中"知三可求     

巧用一个性质解题

<正>等差数列{a_n}的前n项和S n的常用性质很多.在前几年高考试卷中,有几道与等差数列前n项和S_n相关的试题,应用等差数列{a_n}的前n项和S_n的下面的这个性质解决会非常的简便,列举几例与大家分享.基础知识在等差数列{a_n}中,首项a1,公差为d.     

等差数列的性质及其应用

本文介绍等差数列的性质,目的在于掌握等差数列的性质,灵活运用性质解题,以提高解题能力.常用的性质有以下三条:(1)在等差数列{an}中,若m+n=p+q,则am+an=ap+aq(m、n、p、q∈N+).(2)在等差数列{an}中,若m+n=2k,则     

对“an-2+an+2=2an”理解偏差成因分析与思考

<正>一、理解偏差的呈现表达式a_n-2+a_n+2=2a_n(n≥3)是指奇数项成等差数列,偶数项也成等差数列,但这两个等差数列的公差不一定相同.很多同学把上面的表达式错误地理解为a_n+2-a_n=d(n≥1),这个式子同样表示奇数项和偶数项都是等差数列,但这两个等差数列的公差是相同的,都是常数d.     

对等差数列判定方法的研究

一、教学立意 数列是高中数学重要内容之一,有着广泛的实际应用,也是数学研究的重要工具.一方面,数列作为一种特殊的函数与函数密不可分;另一方面,学习数列也为进一步学习数列的极限等内容做好准备.等差数列是一种在数列的学习中有着重要地位的特殊数列.学生在学习等差数列有关概念和性质的基础上,将对等差数列的性质作进一步研究和推广.对等差数列的探索和发现将为今后学习等比数列提供“联想”、“类比”的思想方法.     

走近函数f(x)={x}

题目 (2009年湖北文9)设x∈R,记不超过x的最大整数为[x],令{x}=x-[x],则{√5+1/2},[√5+1/2],√5+1/2 　　A.是等差数列但不是等比数列 　　B.是等比数列但不是等差数列 　　C.既是等差数列又是等比数列 　　D.既不是等差数列也不是等比数列……     

On nonsingular regular magic squares of odd order

Using centroskew matrices, we provide a necessary and sufficient condition for a regular magic square to be nonsingular. Using latin squares and circulant matrices we describe a method of construction of nonsingular regular magic squares of order n where n is an odd prime power.     

用线性取余变换造正交拉丁方和幻方

本文利用线性取余变换造正交拉丁方、幻方和泛对角线幻方。文［１］造奇数阶正交拉丁方的方法，文［２］的方法都本文方法的特例。     

奇数阶泛对角线雪花幻方的构作

The constructional methods of pandiagonal snowflake magic squares of orders 4m are established in paper [3]. In this paper, the constructional methods of pandiagonal snowflake magic squares of odd orders n are established with n = 6m l, 6m 5 and 6m 3, m is an odd positive integer and m is an even positive integer 9|6m 3. It is seen that the number sets for constructing pandiagonal snowflake magic squares can be extended to the matrices with symmetric partial difference in each direction for orders 6m 1 , 6m 5; to the trisection matrices with symmetric partial difference in each direction for order 6m 3.     

On properties of special magic square matrices

By treating regular (or associative), pandiagonal, and most-perfect (MP) magic squares as matrices, we find a number of interesting properties and relationships. In addition, we introduce a new class of quasi-regular (QR) magic squares which includes regular and MP magic squares. These four classes of magic squares are called “special”.We prove that QR magic squares have signed pairs of eigenvalues just as do regular magic squares according to a well-known theorem of Mattingly. This leads to the fact that odd powers of QR magic squares are magic squares which also can be established directly from the QR condition. Since all pandiagonal magic squares of order 4 are MP, they are QR. Also, we show that all pandiagonal magic squares of order 5 are QR but higher-order ones may or may not be. In addition, we prove that odd powers of MP magic squares are MP. A simple proof is given of the known result that natural (or classic) pandiagonal and regular magic squares of singly-even order do not exist.We consider the reflection of a regular magic square about its horizontal or vertical centerline and prove that signed pairs of eigenvalues of the reflected square differ from those of the original square by the factor i. A similar result is found for MP magic squares and a subclass of QR magic squares.The paper begins with mathematical definitions of the special magic squares. Then, a number of useful matrix transformations between them are presented. Next, following a brief summary of the spectral analysis of matrices, the spectra of these special magic squares are considered and the results mentioned above are established. A few numerical examples are presented to illustrate our results.     

Magic square spectra

To study the eigenvalues of low order singular and non-singular magic squares we begin with some aspects of general square matrices. Additional properties follow for general semimagic squares (same row and column sums), with further properties for general magic squares (semimagic with same diagonal sums). Parameterizations of general magic squares for low orders are examined, including factorization of the linesum eigenvalue from the characteristic polynomial.For nth order natural magic squares with matrix elements 1,…,n² we find examples of some remarkably singular cases. All cases of the regular (or associative, or symmetric) type (antipodal pair sum of 1+n²) with n-1 zero eigenvalues have been found in the only complete sets of these squares (in fourth and fifth order). Both the Jordan form and singular value decomposition (SVD) have been useful in this study which examines examples up to 8th order.In fourth order these give examples illustrating a theorem by Mattingly that even order regular magic squares have a zero eigenvalue with odd algebraic multiplicity, m. We find 8 cases with m=3 which have a non-diagonal Jordan form. The regular group of 48 squares is completed by 40 squares with m=1, which are diagonable. A surprise finding is that the eigenvalues of 16 fourth order pandiagonal magic squares alternate between m=1, diagonable, and m=3, non-diagonable, on rotation by π/2. Two 8th order natural magic squares, one regular and the other pandiagonal, are also examined, found to have m=5, and to be diagonable.Mattingly also proved that odd order regular magic squares have a zero eigenvalue with even multiplicity, m=0,2,4,... Analyzing results for natural fifth order magic squares from exact backtracking calculations we find 652 with m=2, and four with m=4. There are also 20, 604 singular seventh order natural ultramagic (simultaneously regular and pandiagonal) squares with m=2, demonstrating that the co-existence of regularity and pandiagonality permits singularity. The singular odd order examples studied are all non-diagonable.     

非素数阶幻方的构造

基于矩阵运算,给出任意双偶数阶和非素数阶幻方的新构造方法:1)由任一低阶m(m为偶数且m≠2)幻方生成一高阶2m阶幻方;2)利用已知的m(m≠2)阶和n(n≠2)阶两个幻方,构造任意的非素数mn阶幻方,加强一些条件后,进一步提出构造两类高级幻方(泛对角线幻方和关联幻方)的新方法.     

Nearly magic rectangles

Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares is magic rectangles which are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to even by odd nearly magic rectangles. We also prove necessary and sufficient conditions for the existence of a nearly magic rectangle, and construct one for each parameter set for which they exist.     

泛系方法论与幻方构造

论述了泛系方法论的精缩影模式及其对求解、建模、算法生成与理论建构的作用,同时用泛系方法提出并证明了:1递归构造n阶幻方(n≥5)的方法;2已知m阶幻方和n阶幻方(m,n≥3),求mn阶幻方的公式;3已知m阶幻方(m≥3),构造2m阶幻方的方法。     

一种4N阶幻方构造方法的论证

一种 4 N阶幻方构造方法被发现 .本文阐明了 4 N阶幻方构造方法 ,介绍了 12阶幻方构造过程 .     

16阶二次幻方的膨胀构造法

2017年詹森构造了6个异基因的8阶二次幻方兼完美幻方,根据它们的特殊性质,创立用一个4阶矩阵代替原有元素的膨胀法,构造出16阶二次幻方兼完美幻方;并对另外2个具有相似性质的8阶二次幻方,也通过膨胀法构造出了16阶二次幻方.