等差数列与等比数列求和公式

　　大家好，今天我们来聊一下等差数列与等比数列求和公式的推导。

　　Derivation of Arithmetic Series Formula 等差数列求和公式的推导

　　When the mathematician Karl Gauss was eight years old, he used the following method to find the sum of the natural numbers from 1 to 100.

　　Let S100 represent the sum of the first 100 numbers. Write out the series in order and then in reverse order.

　　数学家卡尔·高斯八岁时，用下列方法求出1到100之间的自然数之和。

　　让S100表示前100个数字的和。按顺序写出这个系列，然后按相反的顺序写出。

　　The same method can be used to derive the formula for the sum of the general arithmetic series.  For an arithmetic sequence, the terms can be written as:

　　同样的方法也可用于推导一般等差数列之和的公式。对于一个等差数列，这些项可以写成：

　　Where n is the number of terms, a is the first term, d is the common difference between the terms, and tn is the last term.

　　The corresponding arithmetic series is:

　　By substituting  tn=a+n-1d  for tn in formula 1, you can represent the sum of an arithmetic series with a different formula, formula 2:

　　通过用公式1中的tn=a+n-1d替换tn，可以用不同的公式2表示算术级数的和：

　　Formula 2 can be used to determine the sum,  Sn , of the first n terms of an arithmetic series when the value of the last term,  tn , is not known.

　　当最后一项tn的值未知时，公式2可用于确定算术级数的前n项的和Sn。

Derivation of Geometric Series Formula  等比数列求和公式的推导

　　The Canadian Open (also known as Canada Masters) is an outdoor tennis tournament and is one of the major tennis tournaments in the world. In the first round of the men’s singles event, 32 matches are played. In the second round, 16 matches are played. In the third round, 8 matches are played, and so on. These numbers form the terms of a geometric sequence. To determine the total number of matches played in this event, we will need to add these terms together. When we add all the terms of a geometric sequence, the result is called a geometric series.

　　加拿大网球公开赛（又称加拿大大师赛）是一项户外网球比赛，是世界上主要的网球比赛之一。在男子单打第一轮比赛中，共进行了32场比赛。在第二轮比赛中，共进行了16场比赛。第三轮比赛，8场比赛，以此类推。这些数字构成一个等比序列的项。要确定此事件中的比赛总数，我们需要将这些项相加。当我们把一个等比序列的所有项相加时，结果称为一个等比序列。

　　When the terms of a geometric sequence are added, the resulting expression is called a geometric series. The sum,  Sn , of the first n terms of a geometric series is:

　　当等比数列的项被添加时，得到的表达式称为等比数列之和。等比数列的前n项之和为：

　　This can be used to derive a formula for  Sn :

　　Therefore, the sum,  Sn , of the first n terms of a geometric series can be found using the formula:

　　因此，一个等比数列的前n个项的和可以用以下公式求出：

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