Algebraic Formulas

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

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.

• n is the number of items from progression,
• a_k = 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 = a_k - a_k-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 (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, ..., 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

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

where b_k, b_n, b_m, b_p - 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 S_n is determinated by formula

where  b₁ is first item, q - ratio, and b_n - general term of geometric formula.