sn = a 1+a2+……+a(n- 1)+an .

And then through additive commutative law.

sn = an+a(n- 1)+……+a2+a 1 .

Add two types:

2Sn =(a 1+an)+[a2+a(n- 1)]+……+[a(n- 1)+a2]+(an+a 1).

Because in arithmetic progression, a1+an = a2+a (n-1) =1 ...

So 2Sn=n(a 1+an).

So Sn=(a 1+an)*n/2.

Extended data:

Properties of arithmetic series

1. In arithmetic progression, if Sn is the sum of the first n terms of a series, S2n is the sum of the first 2n terms of a series, and S3n is the sum of the first 3n terms of a series, then Sn, S2n-Sn and S3n-S2n are also arithmetic progression.

2. Remember that the sum of the first n terms of the arithmetic series is S..( 1) If a > 0, the tolerance is D.

3. An important condition for a sequence to be arithmetic progression is that the sum of the first n terms of the sequence can be written in the form of S = an 2+BN (where A and B are constants).

Baidu Encyclopedia-arithmetic progression