5: Sequences, Summations, and Logic
In this chapter, we will explore the mathematics behind situations such as these. We will take an in-depth look at annuities. We will also look at the branch of mathematics that would allow us to calculate the number of ways to choose lottery numbers and the probability of winning.
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- 5.0: Introduction to Sequences, Probability and Counting Theory
- A lottery winner has some big decisions to make regarding what to do with the winnings. Buy a villa in Saint Barthélemy? A luxury convertible? A cruise around the world? The likelihood of winning the lottery is slim, but we all love to fantasize about what we could buy with the winnings. One of the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series of regular payments, called an annuity, over the next 30 years or so.
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- 5.1: Sequences and Their Notations
- This section introduces sequences, defining them as ordered lists of terms generated by a specific rule. It covers notations for sequences, including explicit and recursive formulas, and explains how to find terms in a sequence. The section also introduces factorial notation and explores patterns in sequences.
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- 5.2: Arithmetic Sequences
- This section explains arithmetic sequences, where the difference between consecutive terms is constant. It covers explicit and recursive formulas, how to find terms in a sequence, and calculating the sum of an arithmetic sequence. Examples show how to identify and apply these sequences in different contexts.
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- 5.4: Series and Their Notations
- This section explains series as the sum of terms in a sequence and introduces sigma notation for representing series. It covers the distinction between finite and infinite series, including arithmetic and geometric series. Examples show how to calculate sums and apply series in real-world contexts.
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- 5.5: Mathematical Induction
- This section introduces mathematical induction, a proof technique used to establish statements for all natural numbers. It explains the process in two steps: proving a base case and showing that if the statement holds for one integer, it also holds for the next. Examples illustrate how to apply induction to verify formulas and inequalities, providing a foundation for proving mathematical statements systematically.
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- 5.6: Binomial Theorem
- A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find \((x+y)^n\) without multiplying the binomial by itself \(n\) times.
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- 5.7: Chapter Review
- Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain. The probabilities in a probability model must sum to 1. See Example. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment.
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- 5.8: Exercises
- Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain. The probabilities in a probability model must sum to 1. See Example. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment.