Arithmetic progression's general formula is:
an = a 1+(n- 1)d( 1)
The first n terms and formulas are:
Sn=na 1+n(n- 1)d/2 or Sn=n(a 1+an)/2(2).
It can be seen from the formula (1) that 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 (d≠0) or a linear function (d =
Arithmetic average in arithmetic progression: generally set as Ar, Am+an=2Ar, so Ar is the arithmetic average of Am and An.
The relationship between any two am and an is:
an=am+(n-m)d
It can be regarded as arithmetic progression's generalized general term formula.
From arithmetic progression's definition, general term formula and the first n terms formula, we can also deduce that:
a 1+an = a2+an- 1 = a3+an-2 =…= AK+an-k+ 1，k∈{ 1，2，…，n}
If m, n, p, q∈N*, m+n=p+q, then there is.
am+an=ap+aq
Sm- 1=(2n- 1)an，S2n+ 1 =(2n+ 1)an+ 1
Sk, S2k-Sk, S3k-S2k, …, Snk-S(n- 1)k… or arithmetic progression, and so on.
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
Number of items = (last item-first item)/tolerance+1
If a series starts from the second term and the ratio of each term to the previous term is equal to the same non-zero constant, this series is called geometric series. This constant is called the common ratio of geometric series and is usually expressed by the letter q (q≠0). Note: When q= 1, an is a constant series. The general formula of geometric series is: an = a 1 * q (n- 1).
General formula of geometric series
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 the independent variable n, and the point (n, an) is a group of isolated points on the curve y = a 1/q * q x (2) Sum formula: sn = na1(q =1) sn = a1(.
Sum formula of equal ratio sequence
(premise: q≠ 1) the relationship between any two am and an is an = am q (n-m); When using the first N-phase sum of geometric progression, it is important to discuss whether the common ratio Q is 1. (3) We can deduce from the definition of geometric series, the general term formula and the summation formula of the nth term: a1an = a2an-1= a3an-2 = … = akan-k+. If π n = A 1 A2 ... records an an, then π2n- 1=(an)2n- 1, π 2n+1= (an+1) 2n+1. Besides, every item is positive. On the other hand, taking any positive number c as the cardinal number and a arithmetic progression term as the exponent, a power energy is constructed, which is a geometric series. In this sense, we say that a positive geometric series and an arithmetic series are isomorphic. Definition of proportional term: From the second term, every term (except the finite series and the last term) is an equal proportion term of the previous term and the latter term. The middle term formula of equal proportion: An/An- 1=An+ 1/An or (an-1) (an+1) = an2 (5) infinite recursive proportional series summation formula: infinite recursive proportional series summation formula: the absolute value of common ratio is less than 650. (6) The common ratio of the new geometric progression composed of geometric progression: {an} is the common ratio of geometric progression 1 and q if A = A 1+a2+…+anb = an+1+…+a2nc = a2n+1+…… if a. +A4+A7+...+A3n-2b = A2+A5+A8+...+A3n-1c = A3+A6+A9+ ...+A3n, then A, B, A9+.
(1) if m, n, p, q∈N*, m+n=p+q, then am * an = AP * aq(2) In a geometric series, every k term is added in turn, which is still a geometric series. (3) "G is the equal ratio mean of A and B" and "G 2 = AB (G ≠ 0)". (4) If {an} is a geometric series, the common ratio is q 1, {bn} is also a geometric series, and the common ratio is q2, then {a2n}, {A3n}...(5) In the geometric series, the sum of continuous equidistant line segments is equal. (6) If (an) is a geometric series, all terms are positive, and the common ratio is q, then (the logarithm of an based on log) is arithmetic, and the tolerance is the logarithm of q based on log. (7) The sum of the top n terms of geometric progression Sn = A 1 (1-q n)/(1-q) = a1(q n-1)/(q-1).
(1) undetermined coefficient method: it is known that a(n+ 1)=2an+3, a 1= 1, geometric series a (n+1)+x = 2 (an+x) a (. ∫a(n+ 1)= 2an+3 ∴x=3, so (a (n+1)+3)/(an+3) = 2 ∴ {an+3} The first term is 4, and the public ratio is 2.