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Sequences

Sequences

At a Glance - Defining Sequences and Evaluating Terms

Let's imagine, instead of numbers, we have a sequence of animals. The sequence begins with three animals: green tap dancing elephant, purple singing cockatoo, and orange quarterback tiger. We could refer to the first term as green tap dancing elephant, the second as the purple singing cockatoo, and so on. But we'll just become confused in our circus of exotic animal talent.

Or, we could just give each term of the sequence a short label.

We call the first term of a sequence a1 (the elephant in our circus), the second term a2 (the cockatoo), and so on:

a1a2, a3, ..., an, ...

The nth term, an, is called the general term of a sequence.

Sometimes it's useful to think of a sequence as starting with a "zeroth" term instead of a "first" term. Then we would think of the sequence as

a0, a1a2, a3, ..., an, ...

It sounds like we're just being difficult, defining a term by a number that means none. We aren't. We'll tell you why later. For now, assume sequences start with n = 1 unless told otherwise.

With sequences, we like to know how to get there from here. To describe or define a sequence we need to know how to find the general term. It's like knowing how to get from your couch, where you were reading up on Greek mythology for giggles, to the kitchen to fetch a slice of watermelon.

This description may or may not involve a mathematical formula. Some sequences can't be described by a nice tidy formula.

Sample Problem

Define a sequence by

an = high temperature at Shmoop HQ, on the nth day after January 1, 2010.

This is a perfectly reasonable definition. However, as far as anyone knows, there is no mathematical formula that can produce the nth term. Despite our extensive knowledge of rain dancing, we can't predict the weather that accurately.

Even if it's usually 75 and sunny at Shmoop HQ.

Example 1

Define a sequence where

an = the decimal expansion of e to n decimal places.

Find the first four terms of this sequence, starting with n = 1.


Example 2

Evaluate the first four terms of the sequence defined by an = n2, starting with n = 1.


Exercise 1

Describe the sequence

3.1, 3.14, 3.141, 3.1415, 3.14159, ...

What is the general term an?


Exercise 2

Evaluate the first 4 terms of the sequence, starting with n = 1.


Exercise 3

Evaluate the first 4 terms of the sequence, starting with n = 1.

an = 2n + 1


Exercise 4

Evaluate the first 4 terms of the sequence, starting with n = 1.

an = n3


Exercise 5

Evaluate the first 4 terms of the sequence, starting with n = 1.

an = 6 – n


Exercise 6

Evaluate the first 4 terms of the sequence, starting with n = 1.

an = n2 + n


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