怎么求等差数列之和

等差数列是每一项与它的前一项的差等于一个常数的数列。如果要求等差数列之和，你可以将所有数字手动相加。但是，当数列包含大量数字时，就无法使用这种方法了。这时，你可以使用另一种方法，即用数列首项和末项的平均数乘以数列项数，从而快速算出任何等差数列之和。

步骤

评估数列

1. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/a\/a0\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-1-Version-3.jpg\/v4-460px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-1-Version-3.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/a\/a0\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-1-Version-3.jpg\/v4-728px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-1-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 1 确定数列是等差数列。等差数列是一组有规律的数字，其中各数字的增量是一个常数。^[1] 本文所述方法仅适用于等差数列。
   • 要确定数列是否是等差数列，你可以计算前面几个数字之间的差值和最后几个数字之间的差值。等差数列的差值应始终相等。
   • 例如，数列10, 15, 20, 25, 30是一个等差数列，因为各项之间的差值等于常数(5)。
2. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/a\/a7\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-2-Version-3.jpg\/v4-460px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-2-Version-3.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/a\/a7\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-2-Version-3.jpg\/v4-728px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-2-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 2 确定数列的项数。每个数字构成一项。如果数列只包含列出的几个数字，你可以数一数共有多少项。否则，在知道首项、末项，以及被称为公差的各项之差的情况下，你可以使用公式来算出项数。我们可以使用变量${\displaystyle n}$来代表这个数字。
   • 例如，如果你要计算数列10, 15, 20, 25, 30之和，则${\displaystyle n=5}$，因为数列共有5项。
3. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/f\/fe\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-3-Version-3.jpg\/v4-460px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-3-Version-3.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/f\/fe\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-3-Version-3.jpg\/v4-728px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-3-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 3 确定数列的首项和末项。要计算等差数列之和，你必须知道这两个数字。第一个数字常常为1，但也并不一定。我们可以设变量${\displaystyle a_1}$等于数列首项，变量${\displaystyle a_n}$等于数列末项。
   • 例如，在数列10, 15, 20, 25, 30中，${\displaystyle a_1=10}$，而${\displaystyle a_n=30}$。

计算总和

1. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/c\/c3\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-4-Version-3.jpg\/v4-460px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-4-Version-3.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/c\/c3\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-4-Version-3.jpg\/v4-728px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-4-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 1 列出计算等差数列之和的公式。公式为${\displaystyle S_n=n(\frac{a_1+a_n}{2})}$，其中${\displaystyle S_n}$等于数列之和。^[2]
   • 注意，此公式表明等差数列之和等于首项和末项的平均数乘以项数。^[3]
2. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/8\/82\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-5.jpg\/v4-460px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-5.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/8\/82\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-5.jpg\/v4-728px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-5.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 2 将变量$$n{\displaystyle n}、$$a1{\displaystyle a_{1}}和$$an{\displaystyle a_{n}}代入公式中。确保代入步骤正确。
   • 例如，如果数列有5项，首项为10，末项为30，则代入后公式变成：${\displaystyle S_n=5(\frac{10+30}{2})}$。
3. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/c\/c3\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-6.jpg\/v4-460px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-6.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/c\/c3\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-6.jpg\/v4-728px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-6.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 3 计算首项和末项的平均数。将两个数字相加，然后除以2。
   • 例如：
     ${\displaystyle S_n=5(\frac{40}{2})}$
     ${\displaystyle S_n=5(20)}$
4. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/3\/30\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-7.jpg\/v4-460px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-7.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/3\/30\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-7.jpg\/v4-728px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-7.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 4 用平均数乘以数列的项数。这样就算出了等差数列之和。
   • 例如：
     ${\displaystyle S_n=5(20)}$
     ${\displaystyle S_n=100}$
     因此，数列10, 15, 20, 25, 30之和等于100。

完成例题

1. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/1\/16\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-8.jpg\/v4-460px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-8.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/1\/16\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-8.jpg\/v4-728px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-8.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 1 计算1到500之间所有数字之和。考虑所有的连续整数。
   • 确定数列的项数${\displaystyle n}$。由于需要考虑500以内的所有连续整数，因此${\displaystyle n=500}$。
   • 确定数列的首项${\displaystyle a_1}$和末项${\displaystyle a_n}$。由于数列是从1到500，所以${\displaystyle a_1=1}$，而${\displaystyle a_n=500}$。
   • 计算${\displaystyle a_1}$和${\displaystyle a_n}$的平均数：${\displaystyle \frac{1+500}{2}=250.5}$。
   • 用平均数乘以${\displaystyle n}$：${\displaystyle 250.5\times 500=125,250}$。
2. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/c\/c4\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-9.jpg\/v4-460px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-9.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/c\/c4\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-9.jpg\/v4-728px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-9.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 2 求下述等差数列之和。数列的首项为3。数列的末项为24。公差为7。
   • 确定数列的项数${\displaystyle n}$。由于数列的第一项为3，最后一项为24，而每一项比前一项大7，所以这个数列是3, 10, 17, 24。以上推论是根据公差的定义得出，公差即数列中各项与前一项之差。^[4] 这意味着${\displaystyle n=4}$
   • 确定数列的首项${\displaystyle a_1}$和末项${\displaystyle a_n}$。由于数列是从3到24，所以${\displaystyle a_1=3}$，而${\displaystyle a_n=24}$。
   • 计算${\displaystyle a_1}$和${\displaystyle a_n}$的平均数：${\displaystyle \frac{3+24}{2}=13.5}$。
   • 用平均数乘以${\displaystyle n}$：${\displaystyle 13.5\times 4=54}$。
3. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/4\/42\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-10.jpg\/v4-460px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-10.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/4\/42\/Find-the-Sum-of-an-Arithmetic-Sequence-Step-10.jpg\/v4-728px-Find-the-Sum-of-an-Arithmetic-Sequence-Step-10.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 3 解以下问题。陈静在一年的第一周存了5元钱。在这一年中剩下的时间里，她每周会比前一周多存5元钱。年末时，陈静共存了多少钱？
   • 确定数列的项数${\displaystyle n}$。由于陈静存了1年，而1年有52周，所以${\displaystyle n=52}$。
   • 确定数列的首项${\displaystyle a_1}$和末项${\displaystyle a_n}$。她存的第一笔钱金额为5元，所以${\displaystyle a_1=5}$。她在这一年最后一周存的金额可以计算得出，${\displaystyle 5\times 52=260}$。因此，${\displaystyle a_n=260}$。
   • 计算${\displaystyle a_1}$和${\displaystyle a_n}$的平均数：${\displaystyle \frac{5+260}{2}=132.5}$。
   • 用平均数乘以${\displaystyle n}$：${\displaystyle 135.5\times 52=7,046}$。所以，她在年末时共存了7,046元。