高中数列知识梳理
2023-11-23 16:46:06 18 举报AI智能生成
数列是以正整数集（或它的有限子集）为定义域的一列有序的数。数列中的每一个数都叫做这个数列的项。排在第一位的数称为这个数列的第1项（通常也叫做首项），排在第二位的数称为这个数列的第2项，以此类推，排在第n位的数称为这个数列的第n项，通常用an表示。
作者其他创作
大纲/内容
等差与等比
等差数列
通项公式
<span class="equation-text" data-index="0" data-equation="a_n" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="1" data-equation="a_1"><span></span><span></span></span>+(n-1)d
性质
若m+n=p+s
则<span class="equation-text" data-index="0" data-equation="a_m" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="1" data-equation="a_n" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="2" data-equation="a_p" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" contenteditable="false" data-index="3" data-equation="a_s"><span></span><span></span></span>
若m+n=2p
则<span class="equation-text" data-index="0" data-equation="a_m" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="1" data-equation="a_n" contenteditable="false"><span></span><span></span></span>=2<span class="equation-text" data-index="2" data-equation="a_p" contenteditable="false"><span></span><span></span></span>
求和公式的两种形式
<span class="equation-text" data-index="0" data-equation="s_n" contenteditable="false"><span></span><span></span></span>=n(<span class="equation-text" data-index="1" data-equation="a_1" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" contenteditable="false" data-index="2" data-equation="a_n"><span></span><span></span></span>)÷2
<span class="equation-text" data-index="0" data-equation="s_n" contenteditable="false"><span></span><span></span></span>=dn²÷2+n(<span class="equation-text" contenteditable="false" data-index="1" data-equation="a_1"><span></span><span></span></span>-d÷2)
等比数列
通项公式
则<span class="equation-text" data-index="0" data-equation="a_n" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="a_1{q^{n-1}}" contenteditable="false"><span></span><span></span></span>
性质
若m+n=p+s
则<span class="equation-text" data-index="0" data-equation="a_m" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="a_n" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="2" data-equation="a_p" contenteditable="false"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="3" data-equation="a_s"><span></span><span></span></span>
若m+n=2p
则<span class="equation-text" data-index="0" data-equation="a_m" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="a_n" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="2" data-equation="a_p" contenteditable="false"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="3" data-equation="^2"><span></span><span></span></span>
求和公式的两种形式
当q≠1时
<span class="equation-text" contenteditable="false" data-index="0" data-equation="s_n"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\frac{{a_1}(1-q^n)}{1-q}" contenteditable="false"><span></span><span></span></span>
当q=1时
<span class="equation-text" data-index="0" data-equation="s_n" contenteditable="false"><span></span><span></span></span>=n<span class="equation-text" data-index="1" data-equation="a_1" contenteditable="false"><span></span><span></span></span>
数列中常用结论
等差数列
对任意的k∈N*，<span class="equation-text" data-index="0" data-equation="a_1" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="1" data-equation="a_n" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="2" data-equation="a_2" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="3" data-equation="a_{n-1}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="4" data-equation="a_3" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="5" data-equation="a_{n-2}" contenteditable="false"><span></span><span></span></span>=…=<span class="equation-text" data-index="6" data-equation="a_k" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" contenteditable="false" data-index="7" data-equation="a_{n-k+1}"><span></span><span></span></span>。
对任意的k∈N*，有<span class="equation-text" data-index="0" data-equation="s_k" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="1" data-equation="s_{2k}" contenteditable="false"><span></span><span></span></span>-<span class="equation-text" contenteditable="false" data-index="2" data-equation="s_k"><span></span><span></span></span>，<span class="equation-text" data-index="3" data-equation="s_{3k}"><span></span><span></span></span>-<span class="equation-text" data-index="4" data-equation="s_{2k}"><span></span><span></span></span>，…，<span class="equation-text" data-index="5" data-equation="s_{nk}"><span></span><span></span></span>-<span class="equation-text" data-index="6" data-equation="s_{(n-1)k}"><span></span><span></span></span>…成等差数列。
等比数列
对任意的k∈N*，<span class="equation-text" data-index="0" data-equation="a_1" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="a_n" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="2" data-equation="a_2" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="3" data-equation="a_{n-1}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="4" data-equation="a_3" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="5" data-equation="a_{n-2}" contenteditable="false"><span></span><span></span></span>=…=<span class="equation-text" data-index="6" data-equation="a_k" contenteditable="false"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="7" data-equation="a_{n-k+1}"><span></span><span></span></span>。
任意两项<span class="equation-text" data-index="0" data-equation="a_m" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="a_n"><span></span><span></span></span>的关系为<span class="equation-text" data-index="2" data-equation="a_n"><span></span><span></span></span>=<span class="equation-text" data-index="3" data-equation="{{a_m}q^{n-m}}"><span></span></span>
数列求通项
等差数列
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\frac{a_n-a_1}{n-1}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\frac{a_n-a_m}{n-m}" contenteditable="false"><span></span><span></span></span>
等比数列
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\frac{a_n}{a_{n-1}}" contenteditable="false"><span></span><span></span></span>
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