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Numerical Sequences
Numerical SequencesA numerical sequence is a list of numbers arranged in an order. The numerical sequence itself can be either arranged in a particular order or pattern, or arranged without any order or pattern. The elements of a sequence with n element can be arranged in an ordered sequence of natural numbers with the first element equal to 1, i.e. 1, 2, 3, ..., n1, n. The elements of a numerical sequence can therefore be represented by an item name with the corresponding item sequence number, i.e. a1, a2, a3, ..., an1, an. If a numerical sequence is ordered with a pattern, then the elements in the numerical sequence can be expressed in term of the sequence number, n and the term un is usually called a general term of the numerical sequence. For example, the sequence of even natural numbers with sequence number n equal to the number of the element, i.e. a1=2, a2=4, a3=6, ..., an1=2(n1), an=2n. Arithmetical Progression (A.P.)[1]Arithmetical Progression is a numerical sequence of numbers with the difference between two consecutive numbers is equal to a constant number d called common difference. The numerial sequence is of the form a, a+d, a+2d, a+3d, ... . As the arithmetical progression is a numerical sequence with a pattern, the general term of an arithmetical progression can be expressed in term of the item number, The nth term of an arithmetical progression is an=a1+(n1)d with the initial term of an arithmetic progression, a1 and the common difference of an arithmetic progression, d. Or in general, an=ai+(ni)d where ai is the ith term of an arithmetical progression. Arithmetic Series[1]
The sum of an arithmetical progression is called arithmetic series. Therefore an
arithmetic series is of the form S=a+(a+d)+(a+2d)+(a+3d)+.... And the general
format of an arithmetic series is Sn=a1+a2+a3+a4+
....+an1+an =a+(a+d)+(a+2d)+(a+3d)+...+(a+(n2)d)+(a+(n1)d) =∑ n Geometrical Progression (G.P.)[1]Geometrical Progression is a numerical sequence of numbers with the difference between two consecutive numbers is equal to a constant multiplier number r called common ratio. The numerial sequence is of the form a, ar, ar2, ar3, ... . As the geometrcal progression is a numerical sequence with a pattern, the general term of a geometrical progression can be expressed in term of the item number, The nth term of a geometrical progression is an=a1rn1 with the initial term of an geometric progression, a1 and the common ratio of a geometric progression, r. Or in general, an=airni where ai is the ith term of a geometrical progression. Geometric Series[1]
The sum of an
geometrical progression is called geometric series. Therefore an
geometric series is of the form S=a+ar+ar2+ar3+... . And the general format of a geometric series is Snvi8">n=a1+a2+a3+a4+
....+an1+an =a+ar+ar2+ar3+...+arn2+arn1=∑ n If the numerical value of the common ratio, i.e. r or modulus of r, is less than unity, then the sum of the geometric series to infinity can be reduced to Sn=a/(1r). Imply ©sideway ID: 130500018 Last Updated: 5/18/2013 Revision: 0 Ref: References
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