The general formula of arithmetic progression is an = a1+(n-1) d (1).

The first n terms and formulas

The first n terms and formulas are: Sn=na 1+n(n- 1)d/2 or Sn=n(a 1+an)/2 (2).

reason

1. As can be seen from the formula (1), an is a linear function (d≠0) or a constant function (d = 0) of n, and (n, an) is arranged in a straight line. According to formula (2), Sn is a quadratic function of n (d≠0.

2. From the definition and general formula of arithmetic progression, we can also derive the first n terms and formulas: a1+an = a2+an-1= a3+an-2 = … = AK+an-k+1,k ∈ {/kloc-0.

3. if m, n, p, q∈N* and m+n=p+q, then am+an = AP+AQ, sm- 1 = (2n- 1) an, s2n+ 1 = (2n.

If m+n=2p, then am+an=2ap.

4. Other inferences

Sum = (first item+last item) × number of items ÷2

Number of items = (last item-first item) ÷ tolerance+1

First Item =2, Number of Items-Last Item

Last item =2, number of items-first item

The last term = the first term+(number of terms-1)× tolerance.

Inference 3 proves that

If m, n, p, q∈N* and m+n=p+q, then if m, n, p, q∈N* and m+n=p+q, then am+an=ap+aq.

Such as am+an = a1+(m-1) d+a1+(n-1) d.

=2a 1+(m+n-2)d

Similarly,

ap+aq=2a 1+(p+q-2)d

because

m+n = p+q；

A 1, d is a constant.

therefore

If m, n, p, q∈N* and m+n=p+q, then am+an=ap+aq.

Note: 1. Constant series may not be created.

2.m, p, q and n are greater than or equal to natural numbers.

arithmetic mean

Arithmetic average in arithmetic progression: generally set as Ar, Am+an=2Ar, so Ar is the arithmetic average of Am and An, which is the average of series.

The relationship between any two items am and an is: an = am+(n-m) d.

It can be regarded as arithmetic progression's generalized general term formula.