A note On Fascinating Mathematical Applications Arithmetic & Geometric Sequences - Phipher, Angelique; Lemma, Mulatu (IJRDO -JOURNAL OF MATHEMATICS, 2017)
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