Generally speaking, if a series [1] starts from the second term and the ratio of each term to its 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). When applying the first n sums of geometric series [2], we must pay attention to discussing whether the common ratio Q is 1.

In addition, each term is a geometric series with positive numbers, and the same base is taken to form a arithmetic progression; 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, positive geometric series and arithmetic progression are "isomorphic".

Definition of proportional terms: from the second term, every term (except the last term with finite series) is an equal proportion term of the previous term and the latter term.

(1) Infinitely decreasing geometric series summation formula;

Sum formula of infinite recursive equal ratio series: the absolute value of common ratio is less than 1, and the limit when n increases infinitely is called the sum of infinite geometric progression terms.

(2) The norm of a new geometric series composed of geometric series:

{an} is a geometric series whose common ratio is q.

1, if A = A 1+A2+...+ Ann.

Geometric series formula

B=an+ 1+……+a2n

C=a2n+ 1+……a3n

Then, a, b and c form a new geometric series with the common ratio q = q n.

2. if a = A 1+A4+A7+...+A3n-2.

B=a2+a5+a8+……+a3n- 1

C=a3+a6+a9+……+a3n

Then, a, b and c form a new geometric series, and the common ratio Q = Q

2 formula attributes

(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} ... are geometric series, and the common ratios are q12, q13.

(5) In 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 = a1(1-q n)/(1-q) = a1(q n-1)/(q-1).

Note: in the above formula, a n stands for the n power of a.

(8) Because the first term is a 1, the general term formula of geometric series with common ratio q can be written as an = (A 1/q) * Q N, and its exponential function Y = A X is closely related, so we can use the properties of exponential function to study geometric series.

3 General terminology method

1, undetermined coefficient method: it is known that a(n+ 1)=2an+3, a 1= 1, and the geometric series A (n+ 1)+X = 2 (an+X).

a(n+ 1)=2an+x，∫a(n+ 1)= 2an+3 ∴x=3

So (a(n+ 1)+3)/(an+3)=2.

∴{an+3} is a geometric series with a term of 4 and a common ratio of 2, so an+3 = a1* q (n-1) = 4 * 2 (n-1), an = 2 (n+65438

2. definition method: given Sn = A 2 N+B, find the general term formula of an.

∫sn = 2^n+b∴sn- 1=a 2^n- 1+b

∴an=Sn-Sn- 1=a 2^n- 1