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Study Guide

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For those who like pictures better than formulas, we can visualize sequences on number lines and on graphs. For those who like Kit Kats, we can visualize a giant Kit Kat bar. Either way, creating an image will help us understand better how some sequences behave.

Some sequences are well-behaved like well-trained dogs, while others are as unpredictable as wild tigers. If we plot the terms of a sequence on a number line, we can get some intuition for what the terms of the sequence are doing.

Plot the first five terms of the sequence , starting at *n* = 1, on a number line.

Answer.

The first five terms are

Plotting these on a number line, we get

We can see that as *n* gets larger, the terms of the sequence are clumping around 0.

Even if we don't label the terms *a _{n}* on the number line, we can still tell something about what the sequence is doing.

### 2-D Graphs

We have all had sliced bread. It's been around since 1928. Two-dimensional graphs have been around for a while, too. While number lines are nice, we can't tell which dots go with which terms. Since we know as much about 2-D graphs as we do sliced bread, we may as well use them and see what happens.

To graph a sequence on a 2-D graph, we put

*n*on the horizontal axis and*a*on the vertical axis._{n}For each term

*a*of the sequence we graph the point (_{n}*n*,*a*)._{n}### Sample Problem

Plot the first five terms of the sequence , starting at

*n*= 1, on a graph.Answer.

We plot the points

to get this graph:

Even though the terms flip back and forth like a floundering fish, we can see that the values

*a*get closer to 0 as_{n}*n*gets larger.After going through this example, you don't need your fortune-telling turban or crystal ball to see we're heading for limits of sequences. Back the bus up for a second. First, we should go through a couple exercises to see a few more sequences plotted in two dimensions.

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