854 Sequences; Induction; the Binomial Theorem A Look Back, A Look Ahead This chapter is divided into three independent parts: Sections 12.1 —12. 4 , Section 12.5, and Section 12.6. In Chapter 2, we defined a function and its domain, which was usually some set of real numbers. In Sections 12.1—12.4, we discuss a sequence, which is a function whose domain is the set of positive integers. Throughout this text, where it seemed appropriate, we gave proofs of the results. In Section 12.5, a technique for proving theorems involving natural numbers is discussed. In the Appendix Section A.3, there are formulas for expanding ( ) + x a 2 and ( ) + x a . 3 In Section 12.6, we discuss the Binomial Theorem, a formula for the expansion of ( ) + x a , n where n is any positive integer. The topics introduced in this chapter are covered in more detail in courses titled Discrete Mathematics . Applications of these topics are found in the fields of computer science, engineering, business and economics, the social sciences, and the physical and biological sciences. World Population Projected to Reach 9.8 Billion by 2050 The current world population of 7.6 billion is expected to reach 8.6 billion by 2030, 9.8 billion in 2050, and 11.2 billion in 2100, according to a United Nations DESA report titled “World Population Prospects: The 2017 Revision.” Most of the projected increase in the world’s population can be attributed to a short list of high-fertility countries, mainly in Africa and countries that already have large populations. During 2017–2050, half of the world’s population growth is expected to be concentrated in nine countries: India, Nigeria, Democratic Republic of the Congo, Pakistan, Ethiopia, United Republic of Tanzania, United States of America, Uganda, and Indonesia (listed in order of the size of their contribution to the total population growth). Among the ten countries with the largest populations, Nigeria is growing the most rapidly. Currently ranked seventh, Nigeria is projected to become the third largest shortly before 2050. China and India remain the two countries with the largest populations. With more than 1 billion people each, they represent 19% and 18% of the world’s population, respectively. By 2024, the population of India is expected to surpass that of China. Future population growth is highly dependent on the path of future fertility. Relatively small changes in the fertility rate, when projected over decades, can generate large differences in total population. In recent years, the fertility rate has declined in virtually all areas of the world, even in Africa, where fertility levels remain the highest of all major areas. Europe has been an exception to this trend in recent years, with the mean total fertility rate increasing from 1.4 births per woman in 2000 to 1.5 in 2023. Source : Adapted from United Nations Department of Economic and Social Affairs, June 21, 2017, New York —See the Internet-based Chapter Project— Outline 12. 1 Sequences 12. 2 Arithmetic Sequences 12. 3 Geometric Sequences; Geometric Series 12. 4 The Limit of a Sequence; Infinite Series 12. 5 Mathematical Induction 12. 6 The Binomial Theorem Chapter Review Chapter Test Cumulative Review Chapter Project 12 Credit: elenabsl/123RF
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