Order of a certain number of a series called (sequence of number). Each number in the series are called this series entry. Ranked first in the series as the series first one (usually called the first), came in second place is referred to as the series ranked No. 2 ... ... in the first n-bit number called the The first series of n items. Therefore, the general form of the series can be written as a1, a2, a3, ..., an, ... Abbreviated as {an}, the number of a limited number of items listed as "finite number of column" (finite sequence), an infinite number of number of items listed as "infinite series" (infinite sequence). Starting from the first two, each of which is greater than its previous one of the series is called the increment series; Starting from the first two, each of which is less than its previous number of columns is called a descending series; Starting from the first two, and some items is greater than its previous one, some items less than its previous number of columns is called a swing sequence; Cyclical changes in the number of columns is called Cycle series (such as trigonometric functions); The equivalent series called constant column. General formula: number of items out of the first N items in an ordinal number n and the relationship between a formula can be said that this formula is called the sequence of the general formula. Sequence number of the total number of columns in the number of items. In particular, the series can be seen as the positive integer _set_ N * (or its limited sub_set_ {1,2, ..., n}) for the domain of the function an = f (n). If you can use a formula that, it's the general formula is a (n) = f (n).
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Representation
If the sequence {an} of the first n terms and the relationship between the number n can be expressed by a formula, then the formula is called the sequence of general formula. If an = (-1) ^ (n +1) +1 If the sequence {an} of the first n terms with it before one or more of the relationship can be expressed as a formula, then the formula is called a recursive formula of this series. If an = 2a (n-1) +1 (n> 1)
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Arithmetic Sequence
[Definition] In general, if a sequence starting from the first two, each with its differential is equal to the previous one with a constant, this series is called the arithmetic sequence (arithmetic sequence), this constant is called the arithmetic progression of tolerance (common difference), tolerance is usually indicated with the letter d. [Abbreviation] Arithmetic progression can be abbreviated as AP (Arithmetic Progression). [Item] in the arithmetic By three numbers a, A, b consisting of the arithmetic series can be called simple arithmetic progression. At this time, A is called a and b of the arithmetic in the item (arithmetic mean). A relationship: A = (a + b) / 2 [General formula] an = a1 + (n-1) d an = Sn-S (n-1) (n> = 2) [And] the first n items Sn = n (a1 + an) / 2 = n * a1 + n (n-1) d / 2 [Nature] And any two am, an of the relationship: an = am + (nm) d It can be seen as generalized arithmetic progression of the general formula. From the definition of arithmetic sequence, general formula, the first n terms and formulas can be introduced: a1 + an = a2 + an-1 = a3 + an-2 = ... = ak + an-k +1, k ∈ {1,2, ..., n} If m, n, p, q ∈ N *, and m + n = p + q, then there 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 series, and so on. And = (first + last term) × number of items ÷ 2 Number of items = (the last item - the first) ÷ tolerance +1 ÷ = 2 and the first number of items - the last item End items = 2 and ÷ the number of items - the first Let a1, a2, a3 for the arithmetic progression. Items for the arithmetic in the a2, then a2 is equal to 2 times a1 + a3, that 2a2 = a1 + a3. [Application] Everyday life, people often used the arithmetic series such as: the size of the division to the level of various products , When one of the largest size and minimum size or less, often graded according to the arithmetic series. If the arithmetic series, and there is an = m, am = n. then a (m + n) = 0.
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Geometric series
[Definition] In general, if a sequence starting from the first two, each with its ratio is equal to the previous one with a constant, this series is called the geometric series (geometric sequence). This constant is called the geometric series of common ratio (common ratio), the common ratio is usually indicated with the letter q. [Abbreviations] Geometric sequence can be abbreviated as GP (Geometric Progression). [Geometric mean] If a and b into the middle of a number of G, so that a, G, b into a geometric series, then G is called a and b of the geometric mean. A relationship: G ^ 2 = ab; G = ± (ab) ^ (1 / 2) Note: two non-zero real number with the number of geometric mean of two, they are each other's opposite number, so G ^ 2 = ab is a, G, b into a geometric series a few necessary conditions are not sufficient. [General formula] an = a1q ^ (n-1) an = Sn-S (n-1) (n ≥ 2) [And] the first n items When q ≠ 1, the geometric series and the first n terms of the formula Sn = a1 (1-q ^ n) / (1-q) = (a1-an * q) / (1-q) (q ≠ 1) [Nature] Any two am, an of the relationship an = am · q ^ (nm) (3) from the definition of the geometric series, the general formula, the first n terms and formulas can be introduced: a1 · an = a2 · an-1 = a3 · an-2 = ... = ak · an-k +1, k ∈ {1,2, ..., n} (4) geometric mean: aq · ap = ar * 2, ar was ap, aq geometric mean. Remember πn = a1 · a2 ... an, there π2n-1 = (an) 2n-1, π2n +1 = (an +1) 2n +1 In addition, a positive number of both the geometric series after taking the same base composition of a number of arithmetic series; the contrary, to either a positive number C is the end of an arithmetic sequence with the power Can make index construction , is a geometric sequence. In this sense, we say: a geometric series of positive terms with the arithmetic series is the "same structure" of the. Properties: ① If m, n, p, q ∈ N *, and m + n = p + q, then am · an = ap · aq; ② In the geometric series, the k items of each in turn and still into geometric series. "G a, b, the geometric mean" "G ^ 2 = ab (G ≠ 0)". (5) geometric sequence of the first n items and Sn = A1 (1-q ^ n) / (1-q) In the geometric series, the first A1 and common ratio q is not zero. Note: The above formula A ^ n A ^ n said. [Application] Geometric series in life is often applied. Such as: banks have a way --- to pay interest compounded. That the former one count of principal interest Hepburn gold together, Interest in the calculation of the next period, that is, people often say compound interest. Principal and interest and compound interest calculated in accordance with the formula: = principal and interest and principal * (1 + interest rate) ^ term deposit If a sequence starting from the first two, each with its ratio is equal to the previous one with a constant, this series is called the geometric series. This constant is called the geometric series of common ratio, the common ratio is usually expressed with the letter q (q ≠ 0). (1) geometric series of general formula is: An = A1 * q ^ (n-1) If the general formula for the deformation of an = a1 / q * q ^ n (n ∈ N *), when q> 0, the can be seen as an argument to a function of n, the point (n, an) is the curve y = a1 / q * q ^ x on the group of isolated points. (2) summation formula: Sn = nA1 (q = 1) Sn = A1 (1-q ^ n) / (1-q) = (A1-a1q ^ n) / (1-q) = A1 / (1-q)-a1 / (1-q) * q ^ n (ie A-Aq ^ n) (Prerequisite: q is not equal to 1) Any two am, an of the relationship an = am · q ^ (nm) (3) from the definition of geometric series, general formula, the first n items and formulas can be introduced: a1 · an = a2 · an-1 = a3 · an-2 = ... = ak · an-k +1, k ∈ {1,2, ..., n} (4) geometric mean: aq · ap = ar ^ 2, ar was ap, aq geometric mean. Remember πn = a1 · a2 ... an, there π2n-1 = (an) 2n-1, π2n +1 = (an +1) 2n +1 In addition, a positive number of both the geometric series after taking the same base composition of a series of arithmetic; the contrary, to either a positive number C is the end of an arithmetic progression with the power to do index structure Can, is a geometric sequence. In this sense, we say: a geometric series of positive terms with the arithmetic progression is a "homogeneous" of.
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General series of general term demand method
General are: an = Sn-Sn-1 (n ≥ 2) Tired and French (an-an-1 =... an-1 - an-2 =... a2-a1 =... to the above sum can get an). Multiplication by the whole business (the latter for a business with a series containing unknown). Method of return (the number of column deformation, so that the reciprocal of the original series or with a constant and the same as arithmetic or geometric series). Particular: In arithmetic progression, there are always Sn S2n-Sn S3n-S2n 2 (S2n-Sn) = (S3n-S2n) + Sn That the three are the arithmetic series, geometric series in the same. Into three geometric series Fixed point method (the general term commonly used in fractional recurrence relation)
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Special series of general term of the written
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 1,3,5,7,9,11,13,15 .....------- 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) 1,0,1,0,1,0,1,01,0,1,0,1 ....------ 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 1,11,111,1111,11111 .......-------- 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)
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The top N items and a few formulas Method to
(A) 1. Arithmetic progression: General formula an = a1 + (n-1) d first a1, tolerance d, an n-number of items an = ak + (nk) d ak number of entries for the first k If a, A, b constitute the arithmetic progression is A = (a + b) / 2 (2) arithmetic sequence and the first n items: _Set_ of arithmetic series and for the first n terms Sn That Sn = a1 + a2 +...+ an; Then Sn = na1 + n (n-1) d / 2 = Dn ^ 2 (ie n-th power of 2) / 2 + (a1-d / 2) n Have the following summation: 1, 2 induction does not completely reverse cumulative sum method method 3 (B) 1. Geometric series: General formula an = a1 * q ^ (n-1) (ie, q-th power of n-1) a1-led items, an item for the first n an = a1 * q ^ (n-1), am = a1 * q ^ (m-1) Is an / am = q ^ (nm) (1) an = am * q ^ (nm) (2) a, G, b if the composition of geometric mean, then G ^ 2 = ab (a, b, G is not equal to 0) (3) If m + n = p + q is am × an = ap × aq 2 geometric sequence and the first n items Let a1, a2, a3 ... an constitute a geometric sequence The first n items and Sn = a1 + a2 + a3 ... an Sn = a1 + a1 * q + a1 * q ^ 2 +.... a1 * q ^ (n-2) + a1 * q ^ (n-1) (although this formula is the basic formula, but part of the title the first n items and find it is difficult to derive the following that formula, then the basic formula may be derived directly from the past, so I hope that this formula should be understood) Sn = a1 (1-q ^ n) / (1-q) = (a1-an * q) / (1-q); Note: q is not equal to 1; Sn = na1 Note: q = 1 Sum generally have the following five methods: 1, incomplete induction (ie, mathematical induction) 2 3 dislocation multiplication, subtraction tired 4 reverse split item 5 Summation destructive method
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Well-known series
Arithmetic progression typical example: 1 / (1x (1 +1)) +1 / (2x (2 +1)) +1 / (3x (3 +1)) +1 / (4x (4 +1)) +1 / (5x (5 +1 ))............... 1 / (n (n +1)) seeking Sn Resolution: Sn = (1-1/2) + (1/2-1/3) + (1/3-1/4) + (1/4-1/5 )........... .. [1/n-1 / (n +1)] = 1-1 / (n +1) ------ Premium series 0,2,4,8,12,18,24,32,40,50 Through entry type: an = (n × n-1) ÷ 2 (n is odd) an = n × n ÷ 2 (n even) The first n items and formulas: Sn = (n-1) (n +1) (2n +3) ÷ 12 (n is odd) Sn = n (n +2) (2n-1) ÷ 12 (n is even) Premium series from "heaven and earth spectrum", used to explain the principles of Tai Chi derived. Fibonacci sequence 1,1,2,3,5,8,13,21, ... General term type F (n) = (1 / √ 5 )*{[( 1 + √ 5) / 2] ^ n - [(1 - √ 5) / 2] ^ n} This is a completely natural number series, general formula is actually irrational to express. Can also be found in S0 + S1 + S2 + ... ... + Sn-2 = Sn -1
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Encyclopedia
Series progression [Sound of people thrown a few FIl s Shantou class; nporpecc "dish l See arithmetic progression (aritll hoot tic prog it ssion); geometric series (Geolnetric Progression).