Arithmetic progressions
Definitions. The string of numbers (a_{n})_{n}_{ }_{ N} is called arithmetic progression or arithmetic sequence if there is a real number d, called common difference, so
a_{n}_{+1}  a_{n} = 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 a_{n} is called general term of progression or rank n term.
Geometric progressions
Definition. The string of number (b_{n})_{n}_{ }_{ N} is called geometric progression if there is a number q, called ratio, such as
b_{n}_{+1} = b_{n}·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 b_{n} is called general term of rank of the progression with rank n.
Examples: 1, 2, 4, 8, ..., 2^{n}, ... with b_{1} = 1 and q = 2,
5, 15, 45, … with b_{1} = 5 and q = 3.
The term of the geometric progression of rank n determined by formula
b_{n} = b_{1}·q^{n}^{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
b_{k}·b_{n} = b_{m}·b_{p}, 
where b_{k}, b_{n}, b_{m}, b_{p}  terms of a geometric progression b_{1}, b_{2}, ....
The numbers a, b, c build a geometric progression (in this order) iff
b^{2} = ac.
The sum of first n terms of geometric progression S_{n} is determinated by formula

where b_{1} is first item, q  ratio, and b_{n}  general term of geometric formula.
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