常见数列

Matrix posted @ 2008年8月02日 18:49 in Notes with tags 数列 通项公式 前n项和 , 3420 阅读

by F.Yang     

      数列是重要的数学工具, 也是竞赛中的常见考点. 熟悉常见的数列十分重要. 本文整理常见数列的一些资料, 如求和公式, 通项公式等. 由于这些知识比较琐碎, 难免疏漏, 故本文会不定期更新.

1. 等差数列

      通项公式: a_n=a_1+(n-1)d;

      前n项和: S_n=\frac{(a_1+a_n)n}{2}=na_1+\frac{n(n-1)d}{2}.

2. 等比数列

      通项公式: a_n=a_1q^{n-1};

      前n项和: S_n=\frac{a_1(1-q^n)}{1-q}(q\neq1);

                       S_n=na_1,   (q=1);

      无限递降等比数列和: S=a_1/(1-q).

3. Fibonacci 数列

      通项公式: a_n=\frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n];

4.

      1^2+2^2+\cdots+n^2=\frac{1}{6}n(n+1)(2n+1).

5.

      1^3+2^3+\cdots+n^3=\frac{1}{4}n^2(n+1)^2.

 

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