1, general formula: an= a 1+(n- 1)d, where an is the nth term, a 1 is the first term, and d is the tolerance.

2. Sum formula of the first n terms: Sn= n/2*(a 1+an), where Sn is the sum of the first n terms, a 1 is the first term, and an is the nth term.

3. Arithmetic average formula: If A and B are two terms of arithmetic progression, (a+b)/2 is their arithmetic average.

4. Property formula: In arithmetic progression, the product of any two terms is equal to the sum of constants multiplied by their serial numbers. That is, if I and J are two unequal positive integers, and there is no other number between I and J, then AI * AJ = (I+J) * D.

5. Gauss formula: For any real number x, in arithmetic progression, the number of terms not exceeding x is [(x-a 1)/d]+ 1.

6. The sum of the two arithmetic progression counterparts is still arithmetic progression.

7. arithmetic progression's summation formula: Sn= n/2*(a 1+an), where Sn is the sum of the first n terms, a 1 is the first term and an is the nth term.

8. arithmetic progression's term number formula: term number n=(an- a 1)/d+ 1, where an is the nth term, a 1 is the first term and d is the tolerance.

9. arithmetic progression's tolerance formula: d=(an- a 1)/(n- 1), where an is the nth term, a 1 is the first term and d is the tolerance.

10, the relationship between arithmetic progression general term and the first term and tolerance: an= a 1+(n- 1)*d, where an is the nth term, a 1 is the first term and d is the tolerance.

The use of arithmetic series:

1. Calculating mathematical expectation: In probability theory and statistics, arithmetic progression can be used to calculate mathematical expectation. For example, when calculating the average value of a set of data, arithmetic progression's summation formula can be used for quick calculation.

2. Solving geometric problems: In geometry, arithmetic progression can be used to solve some problems related to length and angle. For example, when calculating the bisector of the distance between two points, we can use the properties of arithmetic progression to solve it.

3. Predicting future trends: In finance and economics, arithmetic series can be used to predict future trends. For example, in stock analysis, the concept of arithmetic series can be used to study the historical price trend of stocks and calculate their possible future trends.

4. Scheduling and logistics management: In scheduling and logistics management, arithmetic series can be used to optimize resource allocation and transportation routes. For example, when calculating the distance and time of truck transportation, we can make use of the nature of arithmetic progression to formulate the optimal transportation scheme.

5. Cryptography and coding: In cryptography and coding, arithmetic series can be used to construct some complex passwords and codes. For example, in RSA encryption algorithm, the attribute of arithmetic series can be used to encrypt and decrypt data.

6. Digital signal processing: In digital signal processing, arithmetic progression can be used to represent some simple signals, such as sine wave and cosine wave. Using the properties of arithmetic progression, the signal can be filtered and denoised.