A sequence of numbers arranged in a certain order is called a sequence. Every number in a series is called an item in this series. The number one is called the 1 item of this series (usually also called the first item), the number two is called the second item of this series ... and the number n is called the nth item of this series. Therefore, the general form of the sequence can be written as

a 1，a2，a3，…，an，…

Abbreviated as {an}, a series with limited terms is a "fine series" and a series with infinite terms is an "infinite series".

Starting from the second item, a series with an item greater than the previous item is called an increasing series;

Starting from the second item, the series with each item less than its previous item is called decreasing series;

A series with equal terms is called a constant series; Starting from the second item, some items are larger than the previous item, and some items are smaller than the previous item, which is called wobble sequence;

A series with periodic changes is called a periodic series (such as trigonometric function);

A series with equal terms is called a constant series.

General term formula: The relationship between the nth an of a series and the ordinal n of this series can be expressed by a formula, which is called the general term formula of this series.

The total number of numbers in a series is the number of items in the series. In particular, the sequence can be regarded as a function an=f(n) whose domain is a set of positive integers N* (or its finite subset {1, 2, ..., n}).

If it can be expressed by a formula, its general formula is a(n)=f(n).

Representation method

If the relationship between the nth term of the series {an} and the serial number n can be expressed by a formula, then this formula is called the general term formula of this series. For example, an = (-1) (n+1)+1.

If the relationship between the nth term of series {an} and its previous term or terms can be expressed by a formula, then this formula is called the recurrence formula of this series. For example, an = 2a (n-1)+1(n >; 1)

arithmetic series

definition

Generally speaking, if a series starts from the second term, the difference between each term and its previous term is equal to the same constant. This series is called arithmetic progression, and this constant is called arithmetic progression's tolerance zone. The tolerance is usually expressed by the letter D.

abbreviate

A.p. (arithmetic progression can be abbreviated as A.P.).

arithmetic mean

Arithmetic progression, which consists of three numbers A, A and B, can be called the simplest arithmetic progression. At this time, a is called the arithmetic average of a and B.

General term formula

an=a 1+(n- 1)d

Sum of the first n terms

sn = n(a 1+an)/2 = n * a 1+n(n- 1)d/2

nature

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

App application

In daily life, people often use arithmetic progression, for example, to grade the sizes of various products.

When there is little difference between the maximum size and the minimum size, it is often classified by arithmetic progression.

If it is arithmetic progression, and an = m and am = n, then a (m+n) = 0.

geometric series

definition

Generally speaking, if a series starts from the second term and the ratio of each term to its 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.

abbreviate

Geometric series can be abbreviated as G.P. (geometric series).

geometric mean

If a number G is inserted between A and B to make A, G and B geometric series, then G is called the equal ratio median of A and B. ..

General term formula

an=a 1q^(n- 1)

Sum of the first n terms

When q≠ 1, the formula of the sum of the first n terms of the geometric series is

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

nature

The relationship between any two terms 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 median term: aq ap = ar * 2, ar is ap, and aq is equal ratio median 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 top n terms of geometric progression Sn = a1(1-q n)/(1-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.

App application

Geometric series are often used in life.

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

That is, the previous interest and Hepburn gold price are counted as principal.

In calculating 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.

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 expressed by the letter q (q≠0).

The general formula of 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) Sum formula: Sn=nA 1(q= 1)

sn=a 1( 1-q^n)/( 1-q)

=(a 1-a 1q^n)/( 1-q)

= a1/(1-q)-a1/(1-q) * q n (that is, a-AQ n)

(Premise: Q is not equal to 1)

The relationship between any two terms 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 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.

General term solution of general sequence

Generally speaking, there are:

an=Sn-Sn- 1

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).

Reader's notes

In arithmetic progression, there is always Sn S2n-n S3n-2n.

2S2n-n=(S3n-S2n)Sn

That is, the three are geometric series and geometric series. Third, do arithmetic series.

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 four methods for summation: 1, incomplete induction, and 2 times 3 dislocation summation.