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Progressions Arithmetic ProgressionAn arithmetic progression is a sequence in which each term after the first is formed by adding a fixed amount, called the common difference, to the preceding term. If a is the first term, d is the common difference, and n is the number of terms an arithmetic progression, the successive terms are a, a + d, a + 2d, a + 3d,….a + (n1)d Thus, the last term (or nth term) l is given by l= a + (n  1)d The sum S of the n terms of this progression is given by or Arithmetic
Means
The terms
between the first and last terms of an arithmetic progression are called
arithmetic means between these two terms. Thus to insert k arithmetic
means between two numbers is to form an arithmetic progression of (k+2)
terms having the two given numbers as the first and the last terms.
Geometric
Progression
A
geometric progression is a sequence in which each term after the first is
formed by multiplying the preceding term by a fixed number, called the
common ratio.
If a is
the first term, r is the common ratio, and n is the number of terms, the
geometric progression is:
a, a∙r, a∙r^{2}, ….. a∙r^{n1 }
Thus, the
last ( or nth ) term l is given by l = a∙r^{n1}.
The sum S
of the first n terms of the geometric progression is given by:
or
Geometric
Means
The terms
between the first and the last terms of a geometric progression are called
geometric means between two terms. Thus, to insert k geometric means
between two numbers is to form a geometric progression of k+2 terms having
the two given numbers as the first and last terms.

"Chance favors the prepared mind."  Louis Pasteur 