Over the millenia, legends have developed around mathematical problems involving series and sequences. One of the most famous legends about series concerns the invention of chess. According to the legend, an Indian king summoned the inventor and suggested that he choose the award for the creation of an interesting and wise game. The king was amazed by the “modest” request from the inventor who asked to give him for the first cell of the chessboard 1 grain of wheat, for the second—2 grains, for the third—4 grains, for the fourth—twice as much as in the previous cell, etc. As a result, the total number of grains per 64 cells of the chessboard would be so huge that the king would have to plant it everywhere on the entire surface of the Earth including the space of the oceans, mountains, and deserts and even then would not have enough!
Have you ever thought of how archeologists in the movies, such as Indiana Jones, can predict the age of different artifacts? Do not you know that the age of artifacts in real life can be established by the amount of the radioactive isotope of Carbon 14 in the artifact? Carbon 14 has a very long half-lifetime which means that each half-lifetime of 5730 years or so, the amount of the isotope is reduced by half. Hence, these consecutive amounts of Carbon 14 are the terms of a decreasing geometric progression with common ratio of ½.
This chapter is for those who want to see applications of arithmetic and geometric progressions to real life. There are many applications for sciences, business, personal finance, and even for health, but most people are unaware of these. We will familiarize you with these by giving you five mini-projects and some related problems associated with the concepts afterwards.