The derivation process of summation formula of arithmetic sequence;
Let the first term be a 1, the last term be an, the number of terms be n, the tolerance be d, and the sum of the first n terms be Sn, then sn = (a1+an) n/2; Sn = na1+n (n-1) d/2 (d is the tolerance)
When d≠0, Sn is a quadratic function of n, and (n, Sn) is a set of isolated points on the image of quadratic function. Using its geometric meaning, we can find the maximum value of the first n terms and Sn.
Note: Formulas 1, 2 and 3 are actually equivalent, and the tolerance in Formula 1 is not necessarily required to be equal to one.
It is proved by summation that Sn=a 1+a2+a3+. . . +an①
Sn=an+a(n- 1)+a(n-2)+.。 . +a 1②
①+②: 2sn = [a1+an]+[a2+a (n-1)]+[a3+a (n-2)]+...+[a1+an] (when n is even).
sn = {[a 1+an]+[a2+a(n- 1)]+[a3+a(n-2)]+...+[a 1+an]}/2
Sn=n(A 1+An)/2 (a 1, An, which can be expressed in the form of A 1+(N- 1) D, we can find that the numbers in brackets are all fixed values, that is, (a1+).
Extended reading: five basic formulas of geometric series
The general formula of (1) geometric series is:
An=A 1×q^(n- 1)
If the general formula is converted into an = a 1/q * q n (n ∈ n *), when q >；; 0, an can be regarded as a function of independent variable n, and point (n, an) is a set of isolated points on the curve y = a 1/q * q x.
(2) the relationship between any two am and an is an = am q (n-m).
(3) From the definition of geometric series, the general term formula and the first n terms formula, we can deduce that:
a 1 an = a2 an- 1 = a3 an-2 =…= AK an-k+ 1，k∈{ 1，2，…，n}
(4) Equal ratio mean term: AQAP = Ar 2, Ar is AP, and AQ is equal ratio mean term.
(5) Equal ratio summation: Sn = A 1+A2+A3+...+ Amp.
① when q≠ 1, sn = a1(1-q n)/(1-q) or sn = (a1-an× q) ÷ (/kloc)
② when q= 1, Sn=n×a 1(q= 1).
If π n = A 1 A2 … an, then π2n- 1=(an)2n- 1, π 2n+1= (an+1) 2n+1.