由Sn=f（an）求an的策略

在有关数列问题中，经常要求数列的通项，许多同学对此类问题感到困难．特别是给出Sn与an的函数关系，即Sn=F(an)型，其中Sn表示数列{an}的前n项和，an表示数列的第n项．此类题难就难在关系复杂，不便转化．下面笔者根据自己的教学实践谈一谈此类问题的解题策略．     

由Sn求an

已知一个数列的前n项和Sn，求数列的通项公式an，可由Sn与an的基本关系式 an={S1（n=1）,Sn-Sn-1（n≥2）     

等差数列的一个性质及应用

定理 设数列{an}是以d为公差的等差数列，Sn为{an}的前n项和，记bn=Sn/n，则数列{bn}是以d/2为公差的等差数列．     

思维破定势，反面更简捷，不用“数归法”——简析两道高三数列综合题的解法

题1 数列{an}中，a1=1，当n≥2时，-1/√n-1〈an〈0，Sn为数列前n项的和，且Sn=1/2[an-1/n（n-1）an],（1）求S1，S2，S3，S4的值；（2）求数列{Sn}的通项公式；（3）求limn→∞．an.     

一道高考数列递推题的探究

<正>一、试题呈现(2014年广东高考题)设数列{an}的前n和为Sn,满足Sn=2nan+1-3n2-4n,n∈N*,且S3=15.(1)求a1,a2,a3的值;(2)求数列{an}的通项公式.本题通过Sn和an+1构造了一个递推关系,通过消去Sn,可将本题转化为求一阶线性递推数列的通项问题.但本题所得到的线性递推数列与我们日常所遇到的递推关系有所不同,巧妙在于这是系数为变量的线性递推关系.部分学生遇到此题时发现考题与解题     

构造法在某类特殊数列中的应用

构造辅助数列，求某一类数列的通项an，前n项和Sn及前n项积Ⅱn。     

从一道高考题谈“放缩”的策路

先看2004年一道高考数学题：已知数列{an}的前n项和Sn满足Sn=2an＋（-1）^n（n≥1）．（1）写出数列{an}的前三项a1，a2，a3；（2）求数列{an}的通项公式；     

江苏卷第20题

题目 设M为部分正整数组成的集合,数列{an）的首项a1=1,前n项的和为Sn,已知对任意整数k∈M,当n〉k时,Sn＋k＋Sn-k=2（Sn＋Sk）都成立．     

混合数列的求解方法

我们把关系式中同时含Sn与an的数列,称为混合数列,这类数列的求解方法,就是利用an=Sn-Sn-1（n≥2）消去Sn,或消去an,化为只含通项an类的关系式,或只含和式Sn类的关系式,然后求出an,或求出Sn。     

2011年江苏高考压轴题别解

<正>2011江苏高考第20题:设M为部分正整数组成的集合,数列{an}的首项a1=1,前n项的和为Sn.已知对任意的整数k∈M,当整数n>k时,Sn+k+Sn-k=2(Sn+Sk)都成立.(1)设M={1},a2=2,求a5的值;(2)设M={3,4},求数列     

浅析数列求和方法

数列求和问题有时比较麻烦,甚至无从下手。抓住数列不同的特点,找出规律就可以比较容易地求出来。根据数列的不同特点,给出数列求和的两种方法——添因子求和法和去因子求和法。     

再论一类数列的求和

在求某些离散型随机变量的数学期望时,我们会遇到一类数列的求和问题.将这类问题归纳、拓展为由两个等差数列对应项之积组成的数列的前n项的求和问题及两个等差数列、一个等比数列对应项之积组成的数列的求和问题.利用求和符号及初等数学方法得到这类问题的处理方法、结论及应用的实例.     

运用m序列测量脉冲响应分析

m序列作为脉冲响应测量的激励信号已经被广为接受.除高斯噪声外,实际系统中还存在有色噪声,本文分析了有色噪声干扰下m序列测量技术的统计特性,推出m序列测量方法是无偏的且均方误差和与观测样本数及m序列周期有关的结论;同时,系统中的非线性也影响测量的精确性,分析了系统中存在微弱无记忆非线性时m序列测量方法对非线性失真的抑制能力,并通过Matlab仿真实验进行了验证.     

两个随机变量和的分布类型分析

在三种不同情况下,对两个随机变量和的分布类型进行了研究,证明了一个离散型随机变量与一个连续型随机变量的和在一定条件下为连续型随机变量,发现两个连续型随机变量的和却不一定是连续型随机变量,而两个非连续型随机变量的和也可能为连续型随机变量.     

Some determinants of response summation

Response summation in pigeons was examined in four experiments. In Experiment 1, summation was not found with a compound of two visual stimuli on a television screen after they had individually been used for instrumental conditioning. In this experiment, the training and test trials were separated by an interval during which the television screen was dark. Summation was found in Experiment 2 for which the television screen was permanently white during the interval between trials and in the region that was not occupied by the experimental stimuli. These results were replicated using a within-subject design (Experiment 3) and autoshaping (Experiment 4). Experiment 2 also revealed summation with compounds of auditory and visual stimuli, but not with compounds of two auditory stimuli or two diffuse lights. The results can be explained by a variety of theories of learning, if they take account of generalization between the stimuli.     

关于一道硕士研究生入学试题的推广

2004年漳州师范学院硕士研究生入学考试中有一道高等代数试题,是关于实对称阵的所有正特征根之和与其迹所确定的不等式。证明了这个不等式可推广到实矩阵上去,即实矩阵的所有实部为正的特征根之和与其迹也有类似不等式,同时给出了其等号成立的充要条件。     

Child-Directed Speech Produced by Mothers with Symptoms of Depression Fails to Promote Associative Learning in 4-Month-Old Infants

Child-directed (CD) speech segments produced by 20 mothers who varied in self-reported symptoms of depression, recorded during a structured play interaction with their 2- to 6-month-old infants, were used as conditioned stimuli with face reinforcers in a conditioned attention paradigm. After pairings of speech segments and faces, speech segments were assessed for their ability to increase time spent looking at a novel checker-board pattern (summation test) using 225 4-month-old infants of nondepressed mothers. Significant positive summation, an index of associative learning, was obtained in groups of infants tested with speech produced by mothers with comparatively fewer self-reported symptoms of depression (Beck Depression Inventory or BDI < or = 15). However, significant positive summation was not achieved using speech samples produced by mothers with comparatively more symptoms of depression (BDI > 15). These results indicate that the CD speech produced by mothers with symptoms of depression does not promote associative learning in infants.     

Associative learning phenomena in the snail (Helix aspersa): conditioned inhibition

Two experiments using garden snails (Helix aspersa) showed conditioned inhibition using both retardation and summation tests. Conditioned inhibition is a procedure by which a stimulus becomes a predictor of the absence of a relevant event—the unconditioned stimulus (US). Typically, conditioned inhibition consists of pairings between an initially neutral conditioned stimulus, CS₂, and an effective excitatory conditioned stimulus, CS₁, in the absence of the US. Retardation and summation tests are required in order to confirm that CS₂ has acquired inhibitory properties. Conditioned inhibition has previously been found in invertebrates; however, these demonstrations did not use the retardation and summation tests required for an unambiguous demonstration of inhibition, allowing for alternative explanations. The implications of our results for the fields of comparative cognition and invertebrate physiological models of learning are discussed.     

The role of temporal variables in inhibition produced through extinction

In two experiments with rats as subjects, the temporal characteristics of inhibition produced through extinction were investigated. Each experiment established two independent signals for unconditioned stimulus presentation, one trace and one delay. Following initial training, either the trace or the delay conditioned stimulus (CS) was massively extinguished. In Experiment 1, a summation test established that an extinguished delay CS (but not a neutral CS) passed a summation test with a delay, but not with a trace, transfer excitor, and an extinguished trace CS (but not a neutral CS) passed a summation test with a trace, but not with a delay, transfer excitor. In Experiment 2, a retardation test showed retarded behavioral control by an extinguished delay CS when the CS was retrained as a delay CS, but not as a trace CS, and by an extinguished trace CS when the CS was retrained as a trace CS, but not as a delay CS. The results are discussed in terms of contemporary theories of extinction.     

Conditioning history and inhibitory instrumental stimulus control: Independent-groups and within-subjects measures

The effects of excitatory conditioning history on establishing inhibitory stimulus control have been investigated in classical conditioning, but not in the free-operant paradigm. The present experiments address this question within the context of discriminated free-operant avoidance in which rats’ barpressing postponed shock. When a stimulus with only a history of signaling safety was combined, on a summation test, with a stimulus that maintained avoidance, avoidance rate was reduced, on average, by 60%. In comparison, after a stimulus acquired an excitatory free-operant avoidance history, nonreinforcement alone was not adequate to make it a predictable and effective inhibitor of avoidance on a summation test. These results, consistent with the classical conditioning literature, were produced by both between-group (Experiment 1) and within-subject (Experiment 2) comparisons. These findings are discussed in terms of (1) Konorski’s distinction between “primary” and “secondary” inhibitory stimuli, (2) the Rescorla-Wagner model, (3) the potential contribution of the “reinstatement of fear” to the outcome of summation tests, and (4) their implications for assaying the effectiveness of behavior-modification treatments of phobias.