Progressions
Arithmetic progressions
Definition: An arithmetic progression is a sequence of numbers 
in which
each term, starting from 
, is obtained from the previous one by adding a
constant number called the common difference of the progression. It is written
If 
is the first term, 
the -th term (the general term),
the common difference, the number of terms and 
the sum of the
terms, then we have:
(by definition)
, 
, 
Terms equidistant from the ends
In an arithmetic progression the sum of two terms equidistant from the ends is
equal to the sum of the end terms: 
.
If the number of terms is odd, then there is a middle term 
, such that
.
The necessary and sufficient condition for three terms , taken in that
order, to form an arithmetic progression is that 
.
Geometric progressions
Definition: A geometric progression is a sequence of numbers in which
each term, starting from , is obtained from the previous one by
multiplying it by one and the same 
, called the common ratio. It is
written 
If is the first term, the -th term (the general term),
the common ratio, the number of terms and the sum of the
terms, then we have:
(by definition)
, 
, 
, 
Terms equidistant from the ends
In a geometric progression the product of two terms equidistant from the ends is
equal to the product of the end terms: 
.
If the number of terms is odd, then there is a middle term , such that
.
The necessary and sufficient condition for three terms , taken in that
order, to form a geometric progression is that 
.
Note. If the geometric progression is decreasing 
and the number of
terms is infinite, then
.