A series starting from the second term, the difference between each term and its previous term is equal to the same constant, and such a series is called arithmetic progression. This constant is usually recorded as d, which is called arithmetic progression's tolerance.

Generally, let arithmetic progression be a 1, a2, a3, …, an- 1, an. According to the above definition

an-an- 1=d(n≥2)

Arithmetic progression's general formula can be deduced by incomplete induction:

a2=a 1+d

a3=a2+d=a 1+2d

a4=a3+d=a 1+3d

……

an=a 1+(n- 1)d

Let the sum of the first n items in arithmetic progression be Sn, that is, Sn=a 1+a2+a3+…+an is available.

sn = a 1+(a 1+d)+(a 1+2d)+…+[a 1+(n- 1)d]

sn = an+(an-d)+(an-2d)+…+[an-(n- 1)d]

2sn =(a 1+an)+(a 1+an)+…+(a 1+an)(n)= n(a 1+an)

Sn=n(a 1+an)2

Mathematician Gauss did the problem of 1+2+3+…+ 100 when he was a child, that is, to find the sum of 100 before arithmetic progression with a tolerance of 1. The method that Gauss thought of is exactly the same as the formula of the sum of the first n terms in arithmetic progression.

Arithmetic progression is an ancient mathematics subject. For example, as early as 2700 BC, Egyptian mathematics "Reint papyrus" recorded related problems. In the late Babylonian clay tablets, there was also the arithmetic progression problem of dividing things in descending order. The general idea of one of the questions is:

10 brothers divided into 100 two pieces of silver, the eldest brother was the most, and the same amount decreased in turn. Now I know that BaDi got six taels. What is the difference between two adjacent brothers?

In the Sutra of Zhang Qiujian written by China in the 5th century A.D., through five concrete examples, the general steps of seeking tolerance, summation and term number are given respectively. For example, question 23 on the paper (described in modern language):

There is a woman who is not good at knitting, and the number of knitting is decreasing day by day. It is known that she knitted 5 feet on the first day, 1 foot on the last day and * * * for 30 days. How much did she knit?

This is actually a question of finding the total number of terms by knowing the first term, the last term and the number of terms.

Arithmetic progression has a wide range of practical applications. For example, the size of various products is often divided into several grades. When there is little difference between the maximum size and the minimum size, they are often classified according to arithmetic progression, such as the size of shoes.