What are the summation formulas of 1 and arithmetic progression?

Arithmetic progression's formula an=al+(n- 1)d, the first n terms and formula are: Sn=nal+n(n- 1)d/2.

If the tolerance d= 1: Sn=(al+an)n/2, if m+np+g: am+an=ap+aq exists.

If mtn=2p, then: am+an=2ap, and the above n is a positive integer.

Text translation

The value of the nth item an= the first item+(item number-1)× the sum of the n items before the tolerance Sn = the first item+the last item × item number (item number-1) tolerance /2.

Tolerance d=(an-a 1)÷(n- 1)? ; Number of items (last item-first item)-tolerance+1;

When the sequence is odd, the sum of the first n terms = middle term × number of terms. The series is even. Find the sum of the first term and the second term divided by 2 arithmetic mean formula 2an+l=an+an+2 where {an} is arithmetic progression.

Arithmetic progression correlation formula:

Material n = first material+(material number-1)* tolerance.

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

Tolerance = (last item-first item)/(number of items-1)

Derivation of general formula:

a2-a 1 = d:a3-a2 = d； A4-A3 = d...An-A (n- 1) = d, and add the left and right sides of the above formula respectively to get an-al = (n1) * d → an = al+(n-1) * d.

The first n terms and formulas are: Sn=al*n+[n*(n-l)*d]/2.

Sn=[n*(al+an)]/2

Sn=d/2*n2+(a 1-d/2)*n

Note: All the above n are positive integers.