Chapter 5: Sequences, Summations, and Logic
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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.
This decision is often based on many factors, such as tax implications, interest rates, and investment strategies. There are also personal reasons to consider when making the choice, and one can make many arguments for either decision. However, most lottery winners opt for the lump sum.
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.
- 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.
- 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.
- 5.3: Geometric Sequences
- This section explains geometric sequences, where each term is found by multiplying the previous term by a common ratio. It covers explicit and recursive formulas, methods for finding terms, and applications. Examples illustrate how to apply these concepts.
- 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.
- 5.5: Mathematical Induction
- This section explains the principle of mathematical induction, a proof technique used to establish the truth of statements for all natural numbers. It covers the base step and inductive step, providing a structured process to complete an induction proof. Examples demonstrate how to apply induction to sequences, inequalities, and mathematical identities.
- 5.6: Binomial Theorem
- This section introduces the Binomial Theorem, which provides a formula for expanding expressions of the form (a+b)n. It explains how to use binomial coefficients and Pascal’s Triangle to determine the terms of the expansion. Examples demonstrate how to apply the theorem to simplify polynomial expressions and solve problems involving binomial expansions.