# Algebraic Formulas

## Arithmetic progressions and geometric progressions

Arithmetic progressions

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,   ( n N )

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.

AP (arithmetic progressions) are as a1, a2, ..., an or a1 , a1 + r , a1 + 2r , ... , a1 + (n-1)r where:
• n is the number of items from progression,
• ak = a1 + (k - 1)r , for all k between 1 and n, also called general term formula of an arithmetic progression.
• r is ratio(common difference): r = ak - ak-1 is called the recurrence formula.
• The sum of first  n numbers from an finite arithmetic progression can be calculated as: • E.g.: -1 , 2 , 5, 8, ... with r = 3 and a1 = -1 .

Geometric progressions

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,     (n N).

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.

Keywords: algebra, progressions, geometric progressions, arithmetic progression

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