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Like baseball, or any other sport, there are a few rockstar, standout sequences that get some special attention. They're the Babe Ruth's and Nolan Ryan's of mathematical sequences. They'll turn out to be the key players in calculus related issues, so we're going to spend some time looking them over. If there was a calculus hall of fame, we'd induct them.
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.
2, 4, 6, 8, 10, ...
is an arithmetic sequence. To get from one term to the next, we always add 2.
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.
Is the sequence
4, 8, 12, 16, ...
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.
Is the sequence
3, 6, 10, 15, 21, ...
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.
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.
an + 1 = an + d.
This is the same as saying that to get from one term to the previous term we subtract some constant d.
an + 1 – an = 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.
This means we need to know how many steps it takes to get from one term am to another term an.
We already know that an arithmetic sequence is one where the difference between successive terms is constant. The distance from each term is the same. A geometric sequence is a lot like an arithmetic sequence, but it's completely different at the same time. We can think of it as the doppelgänger of the arithmetic sequence, if we like.
In a geometric sequence, the ratio between successive terms is constant. Geometric sequences grow or shrink at the same ratio from one term to the next. If we divide any two consecutive terms, we'll always get the same ratio.
So if we divide the first terms in a geometric sequence of burritos and get a pinto bean, then dividing burrito-terms two and three of the geometric series will give us a pinto bean of the same size. Let's hope we all like pinto beans.
Is the sequence 2, 4, 8, 16, … geometric?
Maybe, maybe not. We'll need to take a peek at the ratios of successive terms to find out:
Since these ratios are all the same, the sequence is as geometric as the day is long. And since we've been sitting in class for what feels like an eternity, the day is very long indeed.
Is the sequence 5, 10, 15, 20, … geometric?
Once again, it's ratio time. Do all the successive terms of our sequence play off the same playbook?
We don't have to go any farther. The ratios are not all the same, so this sequence is not geometric.
Mom always said we should try to keep a positive attitude, but it's okay for the ratio of a geometric series to be negative. The ratio can be anything nonzero.
When the ratio is negative, successive terms change signs, kind of like the flip-flopping snap of a flag waving in the breeze, always pointing one direction and then the other.
Is the sequence 3, -6, 12, -24, … geometric?
Well, let's take a look at the ratios between the successive terms:
Since the ratios are all -2, this is a geometric sequence.
We mentioned before that although an arithmetic sequence is usually defined in terms of subtraction, we can also think of it in terms of addition. To get from one term to the next, you always add the same thing.
Similarly, while a geometric sequence is defined in terms of division, we can also think of it in terms of multiplication. We said that in a geometric sequence, to get from one term to the previous term, you always divide by the same thing. This is the same as saying that to get from one term to the next, you always multiply by the same thing. This "thing" isn't a swamp monster that has come from the deep to scare little children on Halloween. It is called the common ratio of the sequence and is usually denoted r.
As with the arithmetic sequences, we usually start the geometric sequences off at a1. If we know a1 and r, we know everything there is to know about a geometric sequence. We're the masters of the geometric sequence and maybe the swamp monster too.
A geometric sequence has a1 = 4 and r = 2. What are the first four terms of the sequence?
We're given a1, so we're off to a good start. To get the next term (and the next, and the next), we multiply the term we're on by r = 2:
a2 = a1(2) = 8
a3 = a2(2) = 16
a4 = a3(2) = 32
Like eating a piece of cake, we figured out the general formula on our own. To get from a1 to an, we have to multiply by r a total of (n – 1) times. (If it's not quite clear why it's (n – 1) times, go read about steps in arithmetic sequences).
Multiplying by r a total of (n – 1) times is the same thing as multiplying by rn – 1.
If we start at a1 and multiply by rn – 1, we end up at
an = a1rn – 1.
Because mathematicians need to save their writing hand for scribing infinite sequence strings, it's common to abbreviate a1 by a. so you'll probably see this formula written
an = arn – 1.
In everything we do with geometric sequences from here on, the letter a refers to the first term of the sequence.
Now that we have this nice formula, if we know a and r for a geometric sequence, we can easily find any term we like.
Just like an arithmetic sequence, if we're given two consecutive terms of a geometric sequence, we can work backward to find r and then a. We know that r is the ratio of successive terms:
Once we know r, we can use the formula an = arn – 1 to find a.
Just when we thought we had it all figured out, we have a new problem. How do we find r and a if we have two non-consecutive terms am and an? This is no simple arithmetic sequence. Some people might throw their hands up in the air and go eat lunch. We're going to flex our brain muscles instead of our stomachs.
This is no impossible labyrinth to find our way out of. We only need to perform one extra operation to solve this problem, but it requires knowing how many times we multiply by r to get from am to an (assume m < n).
Let's look at our equation for a geometric sequence. The nth term is an = arn – 1, while the mth term is am = arm – 1.
Let's divide these equations:
We can cancel the a out, and we will multiply by , which is just 1 in disguise. Sneaky devil. This gives us:
Rearranging this equation so that we have r all on its lonesome, we get:
So, once we know how many steps we need to take, we have (n – m). We divide an by am, and then we take the (n – m)th root. That gives us r. Then we proceed as before to find a. Does your brain feel bigger yet?
1) Find .
2) Take the (n – m)th root to get r.
3) Use the geometric sequence formula to find a.
Remember that we should always check our answers by using the values of a and r we found in the formula. If we found the right values, we should be able to use the formula to get the terms am and an that we started with.
As with an arithmetic sequence, a geometric sequence is completely determined by two pieces of information. If you know a and r, you know everything there is to know about the sequence. If you know any two terms am and an, you can find r and a, which means you're the master of the geometric sequence.
When we were young, we were all taught addition and subtraction first. Then we endured the endless multiplication tables, and we learned division is multiplication's partner operation.
We did the same with arithmetic and geometric sequences, which differ only in their operations. We went through arithmetic sequences, which are additive. Then we talked about geometric sequences, which are multiplicative. Now we are going to compare them to each other and to a tasty treat: a Kit Kat bar.
An arithmetic sequence is one where the difference between successive terms is constant.
To determine if a sequence is arithmetic, find the difference between successive terms and see if you always get the same thing. This is like breaking off a piece of a Kit Kat one chocolate-covered wafer at a time.
A geometric sequence is one where the ratio between successive terms is constant.
To determine if a sequence is geometric, find the ratio between successive terms and see if you always get the same thing. We want a big Kit Kat bar for this one. If we break one piece off, the next time we want to break off two pieces, and then four the next time, and then eight. a is one and r is 2.