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.

## Numberlines

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.

### Sample Problem

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.

## Practice:

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

The first five terms are 1, 2, 3, 4, and 5. These terms don't appear to be clumping around anything. The terms are increasing and equally spaced. This sequence isn't an untamed tiger. It's more of a predictable, but untamable goo, slowly oozing down a hill. | |

Plot the first five terms of the sequence on a number line (start with *n* = 1). Describe in words how the terms are behaving as *n* gets larger.

*a*_{n} = 2^{n}

Answer

The first five terms are 2, 4, 8, 16, and 32.

As *n* gets larger, the terms increase, getting farther apart from each other.

Plot the first five terms of the sequence on a number line (start with *n* = 1). Describe in words how the terms are behaving as *n* gets larger.

*a*_{n} = (-1)^{n}

Answer

The first five terms are -1, 1, -1, 1, -1. When we plot these on a number line we only see two dots:

The terms bounce back and forth between 1 and -1 forever.

Plot the first five terms of the sequence on a number line (start with *n* = 1). Describe in words how the terms are behaving as *n* gets larger.

Answer

The first five terms are 5, 4.5, 4.333..., 4.25, 4.2.

The terms are getting closer to 4.