Chapter 15
Sequences and Series
The front row of a theater has eighteen seats, and every row behind it holds two more than the row in front. Which row is the first with forty seats? How many seats does the whole hall hold? You could list the rows one by one and add as you go, but a hall with five hundred rows makes that hopeless. Ordered lists that follow a rule like this are called sequences, and their totals are called series. Both of those questions can be answered without listing anything at all.
What You'll Explore
5 lessons.
- Arithmetic Sequences
Money added to a jar at the same rate every month, or a pile of pipes losing one per row as it rises: both change by a fixed amount at each step. Reaching the five hundredth term by adding five hundred times is no plan, so the real goal is a rule that jumps straight to any position.
- Arithmetic Series
Adding the whole numbers from one to a hundred is a long slog, and a story says a schoolboy once produced the answer in seconds. What had he noticed? This lesson moves from listing the terms of a sequence to adding them all, and goes after the observation that turns a long sum into a short one.
- Geometric Sequences
Bacteria that split in two every hour do not grow by a fixed amount, they grow by a fixed multiple, and a car losing a fifth of its value each year shrinks the same way. This lesson asks what changes when each step multiplies rather than adds, and what that multiplier decides about where the list is headed.
- Geometric Series
A king promised one grain of wheat on the first square of a chessboard, and twice as many on each square after as on the one before. How much wheat is that altogether? And can a sum that never stops have a total at all? Both questions are settled here.
- Telescoping Sums
Some sums have no fixed difference and no fixed multiple, so nothing else in this chapter reaches them. Add the first few terms anyway and the running totals fall into a pattern too neat to be an accident. Here you will find where that neatness comes from, and whether it holds when the sum never ends.