Chapter 11
Sequences and Series
Stack cans in a pyramid: 3 on the top row, 4 below it, 5 below that. Anyone can name the next row. Ask what the fortieth row holds, or how many cans the whole stack needs, and counting row by row stops being an answer. A list built by a rule can run far past where you stopped writing, but the rule itself stays short. Sequences and series are the tools for working with the rule instead of the list, and it is surprising how much a short rule can tell you.
What You'll Explore
5 lessons.
- Sequences and Notation
Write down 2, 5, 10, 17 and someone can probably continue it, but nothing in the list itself says which rule you had in mind. You will look at what a list of numbers really is, how a rule pins one down exactly, and how a long sum gets written without writing all of it.
- Arithmetic Sequences and Series
Count by sevens, or build a theater where every row holds two more seats than the one in front. Both lists grow by the same fixed step each time. Finding the hundredth row without listing everything before it is one question; adding all hundred rows is the other.
- Geometric Sequences and Series
Money in an account does not grow by a fixed amount each year, but by a fixed percentage. Lists built by a repeated factor behave nothing like lists built by a repeated step, and adding up a run of their terms looks hopeless at first. Both of those are what this lesson takes on.
- Infinite Geometric Series
Cut a cake in half, then halve what is left, and keep going forever. The pieces you have taken look as though they fill the cake, but addition works on two numbers at a time, and an endless list has no last step. Giving that endless total a meaning comes first.
- The Binomial Theorem
Squaring x plus y is easy and cubing it is manageable, but nobody multiplies out ten identical factors one at a time. The expansion has a predictable shape, and the numbers sitting in front of its terms follow a pattern of their own. Tracking down where that pattern comes from is the work here.