A note On Fascinating Mathematical Applications Arithmetic & Geometric Sequences

Abstract

A sequence is a set of numbers in a specific order. The two simplest sequences
which are interesting to work with are the classical arithmetic and geometric sequences.
Since arithmetic and geometric sequences are so nice and regular, they have simple and
friendly formulas. They have many interesting mathematical properties which are enjoyable
and have exciting mathematical patterns.
In mathematics, an arithmetic sequence is a sequence of numbers such that the difference of
any two successive members of the sequence is a constant called common difference. For
instance, the sequence 3, 5, 7, 9, 11, 13... is an arithmetic sequence with common difference
2.
If the initial term of an arithmetic sequence is A1 and the common difference of successive
members is d, then the nth term A
n
of the sequence is given by:
A
n = A1 + (n - 1)d
Likewise, a geometric sequence is a sequence of numbers where each term after the first is
found by multiplying the previous one by a fixed non-zero number called the common ratio.
For example, the sequence 2, 6, 18, 54, ... is a geometric sequence with common ratio 3.
Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2
If the initial term of a geometric sequence is G1 and the common ratio r, then the nth term
G
n
of the sequence is given by:
Gn = G1 r
(n - 1)
These two sequences have many mathematical properties and patterns that are worthy of
exploration in today's mathematics world.
In this paper we will make our journey with the fascinating mathematical beauty of these two
celebrity sequences. The purpose of the study is to dig out some important results and
practical applications concerning the arithmetic and geometric sequences. We deeply
examine some of the interesting properties and patterns of these two shining stars of theclassical number sequences. Some important theorems dealing with the mathematical
concepts of the two sequences will be proved. We also investigate beautiful connections that
exist between these sequences and the seemingly unrelated mathematical territories of
perfect and triangular numbers. Different impressive problem solving techniques will be
shown. Real life application of arithmetic and geometric sequences will be discussed.
2000 Mathematical Subject Classification: Primary 40B05