If a series starts from the second term and the ratio of each term to the previous term is equal to the same constant, this series is called geometric series. This constant is called the common ratio of geometric series and is usually represented by the letter Q.

The general formula of (1) geometric series is: an = a 1× q (n- 1).

If the general formula is transformed 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 set of isolated points on the curve y = a1/q * q X.

(2) the relationship between any two am and an is an = am q (n-m).

(3) A1an = a2an-1= a3an-2 = … = akan-k+1,k ∈ {1 can be deduced from the definition of geometric series, the general term formula, the first n terms and the formula.

(4) Equal ratio mean term: AQAP = Ar 2, Ar is AP, and AQ is equal ratio mean term.

If π n = A 1 A2 … an, then π2n- 1=(an)2n- 1, π 2n+1= (an+1) 2n+1.

In addition, each term is a geometric series with positive numbers, and the same base number 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, we say that a positive geometric series and an arithmetic series are isomorphic.

Nature:

(1) if m, n, p, q∈N*, and m+n = p+q, then am an = AP AQ；;

(2) In geometric series, every k term is added in turn and still becomes a geometric series.

G is the median term in the equal proportion of A and B, and G 2 = AB (G ≠ 0).

(5) The sum of the first n terms of geometric series Sn = a1(1-q n)/(1-q) or Sn = (a1-an * q)/(q)

In geometric series, the first term A 1 and the common ratio q are not zero.

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

Geometric series are often used in life.

For example, banks have a way of paying interest-compound interest.

That is, the interest and principal of the previous period are added together to calculate the principal.

Then calculate the interest of the next period, which is what people usually call rolling interest.

The formula for calculating the sum of principal and interest according to compound interest: sum of principal and interest = principal *( 1+ interest rate) deposit period.

Arithmetic series formula:

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

Or an=am+(n-m)d

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

If m+n=p+q, then: am+an=ap+aq exists.

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

All the above n are positive integers.

Text translation

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

Sum of the first n items = (first item+last item) * Number of items /2

Tolerance = Last Item-First Item

Symmetric sequence formula:

General formula of symmetric sequence:

The total number of items in a symmetric sequence: represented by the letter S.

Symmetric sequence items: represented by the letter C.

Symmetric sequence tolerance: represented by the letter d.

Common ratio of equal ratio symmetric sequence: expressed by the letter Q.

Let k=(s+ 1)/2.

General term solution of general sequence

Generally speaking, there are:

an=Sn-Sn- 1 (n≥2)

Cumulative sum (an-an-1= ... an-1-an-2 = ... a2-a1= ... add the above items to get one).

Quotient-by-quotient total multiplication (for a series with unknown numbers in the quotient of the latter term and the previous term).

Reduction method (transforming a sequence so that the reciprocal of the original sequence or the sum with a constant is equal to the difference or geometric series).

Special:

In arithmetic progression, there are always Sn S2n-Sn S3n-S2n.

2(S2n-Sn)=(S3n-S2n)+Sn

That is, the three are arithmetic progression and geometric progression. Tri-geometric series.

Fixed point method (often used in general fractional recurrence relation)

How to write the general term of special sequence?

1,2,3,4,5,6,7,8.......- an=n

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8......- an= 1/n

2,4,6,8, 10, 12, 14.......- an=2n

1,3,5,7,9, 1 1, 13, 15.....- an=2n- 1

- 1, 1,- 1, 1,- 1, 1,- 1, 1......an=(- 1)^n

1,- 1, 1,- 1, 1,- 1, 1,- 1, 1......——an=(- 1)^(n+ 1)

1,0, 1,0, 1,0, 1,0 1,0, 1,0, 1....an=[(- 1)^(n+ 1)+ 1]/2

1,0,- 1,0, 1,0,- 1,0, 1,0,- 1,0......- an=cos(n- 1)π/2=sinnπ/2

9,99,999,9999,99999,.........an=( 10^n)- 1

1, 1 1, 1 1 1, 1 1 1 1, 1 1 1 1 1.......an=[( 10^n)- 1]/9

1,4,9, 16,25,36,49,.......an=n^2

1,2,4,8, 16,32......——an=2^(n- 1)

Solution of the summation formula of the first n terms of the sequence

1。 Arithmetic series:

The general formula an=a 1+(n- 1)d, the first term a 1, the tolerance d, the nth term of an.

An=ak+(n-k)d ak is the k th term.

If a, a and b constitute arithmetic progression, then A=(a+b)/2.

2. The sum of the first n items in the arithmetic series:

Let the sum of the first n terms of arithmetic progression be Sn.

That is, Sn=a 1+a2+...+ An;

Then Sn=na 1+n(n- 1)d/2.

= dn 2 (that is, the second power of n) /2+(a 1-d/2)n

There are also the following summation methods: 1, incomplete induction 2, accumulation 3, and inverse addition.

(2) 1. Geometric series:

The general formula an = a 1 * q (n- 1) (that is, n- 1 power of q) is the first term, and an is the nth term.

an=a 1*q^(n- 1),am=a 1*q^(m- 1)

Then an/am = q (n-m)

( 1)an=am*q^(n-m)

(2) If A, G, and B constitute a neutral term with equal proportion, then g 2 = ab (a, B, and G are not equal to 0).

(3) if m+n=p+q, am×an=ap×aq.

2. The first n sums of geometric series

Let a 1, a2, a3 ... a geometric series form.

The sum of the first n terms Sn=a 1+a2+a3 ... one; one

sn = a 1+a 1 * q+a 1 * q 2+...a 1 * q(n-2)+a 1 * q(n- 1)(。

sn=a 1( 1-q^n)/( 1-q)=(a 1-an*q)/( 1-q)；

Note: Q is not equal to1;

Sn=na 1 note: q= 1.

There are generally five methods for summation: 1, complete induction (that is, mathematical induction), 2 multiplication, 3 dislocation subtraction, 4 reverse summation, and 5 split term elimination.