Study Guide

Sequences

Sequences Introduction

Our calculus studies began with encountering a hungry bear. He looked up at us, licked his lips, and charged at us with the fury of a steam-powered locomotive. Being the smart people we are, we ran. What else is there to do? We're not confronting a ravenous bear with nothing but a piece of paper, a pencil, and a graphing calculator.

Despite running, we couldn't outrun the hungry bear. We encountered limits, derivatives, integrals, and even some weird things about counting to infinity. We've managed to wrestle our way free a couple times, and the bear decided to leave us alone. As far as we've come, there's no turning back. But we can take a break for a second and enjoy the scenery.

Now that we've seen the worst calculus has to dole out, there are a couple more concepts we need to understand before we can declare ourselves masters of the calculus circus. The first we already know from second grade. No, we're not talking about delicious peanut butter and potato chip sandwiches. We are talking about patterns.

In calculus, we're going to practice our pattern-finding on lists of numbers, also known as sequences. We'll see that we already know a good deal about sequences, and we'll learn some neat tricks to impress our friends with, sort of like balancing a baseball bat on our chin while singing our ABC's. We can also look for patterns in sets, functions, and the real world, but sequences of numbers are a nice place to start.

Sequences Resources

Websites

Arithmetic and Geometric Sequences
If the names arithmetic and geometric seem vague, that’s because their real identities are veiled in the halls of mathematicians since the time of Archimedes. With this link, you won’t be able to spy in on these top secret halls, but you will gain some insight into what these sequences are all about.

Examples of Convergent and Divergent Sequences
The best way to classify a Sasquatch is by comparing it to other animals you already know. With this link, you can compare new sequences to these to see if they are convergent or divergent.

Examples for Determining Convergence and Divergence
If a black hole is a divergence, does that make a white hole a convergence? Ponder over that while you look at these examples to determine convergence and divergence of sequences.

Videos

Arithmetic and Geometric Sequences
Have you ever wondered what Einstein looked like before his hair turned white? This instructor gives a fairly good attempt at impersonating a young Einstein while teaching about arithmetic and geometric sequences.

How to Find the Equation of an Arithmetic Sequence
You can be the Sherlock Holmes of sequences. You’ll learn how to deduce the equation of arithmetic sequences in this video. It’s not necessarily elementary.

How to Find r in a Geometric Sequence
This link will teach you how to find the ratio that binds a geometric sequence of numbers together.

Showing Convergence and Divergence of Sequences
Is the sequence like a well-behaved group of school children walking single-file across the street, or is it more like a pack of wolves run amok across the countryside? After watching this video, you’ll know.

Games and Tools

Identifying and Understanding Arithmetic and Geometric Sequences
Step right up, step right up. Test your superior sequencing skills. Tell if the sequence is arithmetic or geometric. Tell me what the formula for the nth term is.

How to Calculate Terms of an Arithmetic Sequence
Line up a bunch of M&Ms into a sequence by color. Several M&Ms need to survive being eaten for you to see the sequence. This tool teaches you how to find any term of an arithmetic sequence with just two terms given. Arithmetic sequences > M&M sequences.

Tool to Visualize Sequences
Do sequences feel a lot like leprechauns to you? You know they exist, but you just can’t see them. This tool will help you to visualize any sequence you desire. Look hard, and you may find that pot of gold at the end of the rainbow.