Definitions. The string of numbers (an)n N is called arithmetic progression or arithmetic sequence if there is a real number d, called common difference, so
an+1 - an = d, ( nN )
i.e. if each term of the sequence (starting second) is equal to the previous one plus one and the same number (ratio - common difference).
The item an is called general term of progression or rank n term.
Definition. The string of number (bn)n N is called geometric progression if there is a number q, called ratio, such as
bn+1 = bn·q, (any n N)
i.e. if each term of the sequence (starting second) is equal to the product of the previous term and one and the same number (ratio).
The item bn is called general term of rank of the progression with rank n.
Examples: 1, 2, 4, 8, ..., 2n, ... with b1 = 1 and q = 2,
5, 15, 45, … with b1 = 5 and q = 3.
The term of the geometric progression of rank n determined by formula
bn = b1·qn-1, (nN).
The square term of rank n is equal to the product terms equidistant from it:
in particular case, for any three consecutive terms
If k + n = m + p (k, n, m, p N), then
bk·bn = bm·bp,
where bk, bn, bm, bp - terms of a geometric progression b1, b2, ....
The numbers a, b, c build a geometric progression (in this order) iff
b2 = ac.
The sum of first n terms of geometric progression Sn is determinated by formula
where b1 is first item, q - ratio, and bn - general term of geometric formula.
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