Why is the concept of sequences important in real life What

Why is the concept of sequences important in real life? What is the difference between a finite and an infinite sequence? What is the difference between an arithmetic and geometric sequence? Give the functional definition for each of them. What distinguishes a sequence from a series? Can you think of a real life application of concept of mathematical series from your readings or personal research?

Solution

The sequence of numbers (or functions) in which the first (usually) number is given and every next number (term, element) is defined via prior number (or numbers) by an explicit fomula or a recurrent relationship. The sequence of natural numbers

1.2.3.4.5..........

is the example of an arithmetic series in which each next number can be found by adding 1 to the prior number, and the first number is 1. In the arithmetic series the difference between two consequitive elements is constant, such as

1 3 5 7 9 11 .... 5 15 25 35 45 ..........

In a geometric series the ratio of two consequitive elements is constant:

1 3 9 27 81 ........ 1/2 1/4 1/8 1/16 1/32 ........

The have a lot of applications in real life, physics, calculus. Some integrals in calculus can be calculated with the use of infinte arithmetic series. It helps to test integration formulas if you are doing this first time