Arithmetic Sequences

An arithmetic sequence is a sequence where the step from one term to the next is constant. That is, you always add the same thing to get from one term to the next.

An arithmetic sequence is like going up a huge flight of stairs. Most days, you'll walk up the flight of stairs one step at time. Sometimes, though, you are in a hurry, and you skip stairs as you rush up the stairs to get your coat before the bus comes. The increment between each of your steps is constant: one stair if you are on time, and two stairs if you are in a hurry. Be careful not to trip. A bruised shin is the worst.

Sample Problem

The sequence

2, 4, 6, 8, 10, ...

is an arithmetic sequence. To get from one term to the next, we always add 2.

Sample Problem

The sequence

2, 4, 8, 16, ...

is not an arithmetic sequence. To get from 2 to 4 we add 2, but to get from 4 to 8 we add 4.

Since we didn't add the same thing every time, this isn't arithmetic.

Saying the step up from one term to the next term is constant is the same as saying the step down from one term to the previous term is constant. In an arithmetic sequence, the difference between one term and the previous term must always be the same. Our base-running analogy breaks down here. You wouldn't want to run them backwards unless you go from first to second base facing the wrong way.

Sample Problem

Is the sequence

4, 8, 12, 16, ...

arithmetic?

Answer.

Let's find the difference between each pair of successive terms.

8 – 4 = 4

12 – 8 = 4

16 – 12 = 4

The difference between each pair of consecutive terms is 4.

That means this is an arithmetic sequence.

Sample Problem

Is the sequence

3, 6, 10, 15, 21, ...

arithmetic?

Answer.

Let's find the difference between each pair of terms.

6 – 3 = 3

10 – 6 = 4

We don't have to go any further. The difference between successive terms isn't always the same. That means this is not an arithmetic sequence.

It's also okay for the step from one term to the next to be negative, as long as the step is constant.

Sample Problem

The sequence

10, 0, -10, -20, ...

is an arithmetic sequence because the step from one term to the next is always -10.

Now that we're clear on what an arithmetic sequence is, let's put the definition into symbols and an equation. Mathematicians love equations.

In an arithmetic sequence, to get from one term to the next term we add some constant d.

In symbols,

a_{n + 1} = a_n + d.

This is the same as saying that to get from one term to the previous term we subtract some constant d.

In symbols,

a_{n + 1} – a_n = d.

Arithmetic sequences are usually defined in terms of subtraction rather than addition. The value d is called the common difference for the sequence.

We'll start all our arithmetic sequences with n = 1 corresponding to the first term, just because we can. We can also jump up on our desks and sing, "Take Me Out to the Ball Game," at the top of our lungs, but it probably won't help us with sequences.

An arithmetic sequence is completely determined by two things: its starting term a₁ and its common difference d. Once you know where an arithmetic sequence starts and what its step size is, you know everything there is to know about it.

If we throw a baseball backwards behind us, we might break our mother's chandelier. But if we know two terms in an arithmetic sequence, we can work backwards to figure out a₁ and d.

We can figure out a₁ and d even if we're given two terms a_m and a_n that aren't consecutive (assume m < n). We have to do one extra operation. After finding the difference a_n – a_m, we have to divide by the number of steps required to get from a_m to a_n. That gives us d, and we can proceed as before.

This means we need to know how many steps it takes to get from one term a_m to another term a_n.