1. Definition: If the difference between each item of a series and its previous item is equal to the same constant of the second item, the series is called arithmetic progression. The symbol is an+ 1-an=d(n∈N*, d is a constant).

2. Arithmetic average term: The necessary and sufficient condition for the series A, A and B to become arithmetic progression is A=(a+b)/2, where A is called the arithmetic average term of A and B. 。

Second, arithmetic progression's related formulas.

1. general formula: an = a1+(n-1) D.

2. The first n terms and formulas: sn = na1+n (n-1)/2d+d = (a1+an) n/2.

Third, the nature of arithmetic progression

1. if m, n, p, q∈N*, m+n=p+q, {an} is arithmetic progression, then am+an=ap+aq.

2. In arithmetic progression {an}, ak, a2k, a3k, a4k, ... are still arithmetic progression with an error of kd.

3. If {an} is arithmetic progression, then Sn, S2n-Sn, S3n-S2n, ... are still arithmetic progression with an error of n2d.

4. Increase or decrease of arithmetic progression: d>0 is an increasing sequence, when a 1

5. The first term of arithmetic progression {an} is a 1 with a tolerance of d. If the sum of the first n terms can be written as Sn=An2+Bn, then A=d/2 and B=a 1-d/2. When d≠0 represents a quadratic function, the first n of the sequence {an}.

Fourth, the solution to the problem

1. Three kinds of problems related to the sum of the first n terms

(1) If you know any three of a 1, D, N, an and Sn, you can get the other two, which embodies the idea of the equation.

(2)Sn = d/2 * N2+(a 1-d/2)n = An2+Bn？ d=2A。

(3) When the image of quadratic function is used to determine the maximum value of Sn, the ordinate of the highest point is not necessarily the maximum value, and the ordinate of the lowest point is not necessarily the minimum value.

2. Setting elements and solving skills

It is known that three or four numbers form a kind of arithmetic progression, so you should be good at setting elements. If the odd number becomes arithmetic progression and the sum is constant, it can be set to …, a-2d, a-d, a+d, a+2d, …;

If the even number is arithmetic progression and the sum is constant, it can be set as …, a-3d, a-d, a+d, a+3d, …, and the rest items are set symmetrically according to arithmetic progression's definition.