Geometric Series

Objective

• Review over Arithmetic Sequences
• Review over Geometric Sequences
• Go over the importance of the ratio and the first term
• Go over how to solve the Geometric Series to the nth term
• Do some examples
• Homework

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OKAY, SO THEN WHAT’S A SEQUENCE?

A sequence is just an ordered list of numbers.

Something like:

Something important to note, each number in the list is called a term of the sequence.

So, for example, the first term of this sequence is 4.

Now we care about this because it allows us to take this basic sequence, and turn it into a function.

For example, we can see that as we move up the sequence

We are adding 3 to the term before.

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+3

+3

+3

+3

+3

+3

+3

This is what we would call a recursive pattern.

Determining the Recursive Pattern

So how do we actually take a recursive pattern and determine it as a function?

Well, to be honest, we take what we are given, and plug them in for x and y.

It sounds worse than it is, so let’s do an example:

Example 1:

Convert the following sequence to a function:

To start, let’s look at this sequence in terms of x and y:

Now, that we can see our sequence in a way we can recognize, we need to figure out what this looks like as a function.

First, let’s see what recursive pattern we can find in this sequence.

So, looking at the y’s, we can see that when x = 0, y = 5.

So, our function is going to look something like:

Now we need to look at the pattern we can see from the sequence.

As we can see, the sequence is adding 2 to each of the y’s.

This actually translates as multiplying the x by 2.

Like so:

And now, we convert back to function notation:

And there is our function!

Explicit versus Recursive

So now that we know of a way to determine a function when given just a sequence.

We need to tell the difference between an explicit rule, and a recursive rule.

An explicit rule defines the term of n as a function of n, where as a recursive rule defines the term in position n by relating it to one or more previous terms.

So for example, using our last sequence:

Now we know, from what we saw before, our function is:

But, since this is the function that defines the term n, as a function of n, this is the explicit rule.

The recursive rule deals with former terms.

So, we can see looking at previous terms, that we are just adding 2 each time.

So the recursive rule would look something like:

So, how do we know this works?

Well because if we try out what it says for, let’s say, the 4^th term:

So, the difference between the explicit and recursive rule, is with the explicit rule, you don’t need the sequence to find it.

For the recursive, you have to have the sequence to find it.

Let’s do a few more examples so you can get the picture:

Example 2:

Determine the explicit and recursive rules for this sequence.

First things first, let’s find the pattern:

We can see that we are adding 7 each time

So, recursive rule is going to be:

Now we need to find our explicit rule.

We know we’re adding 7 each time

And that adding translates to multiplying

So, our explicit rule is going to look something like:

Now, we plug in our n, and f(n) to see what we need to add/subtract.

So, let’s start with n = 5

Like so:

Now we need to figure out what c is.

To do that, we need to solve for c:

-35 -35

So, our explicit rule is:

SO THEN, WHAT’S A GEOMETRIC SEQUENCE?

Well, a geometric sequence deals with exponential functions instead of linear ones.

Basically, where an arithmetic sequence deals with adding to the former (recursive)

The geometric sequence deals with multiplying to the former (still recursive)

So, to figure out what to do, we need to use the method of finite differences.

THE METHOD OF FINITE DIFFERENCES

To use the method of finite differences, we need to first start with a sequence first:

Next, we need to add “levels” to it to figure out what the power is:

The way we use the levels, is we figure out what we need to add to the sequence to get the next number.

For example:

+2

+6

+18

+54

Now, since we didn’t find a pattern, we go to the next level:

Looking at level 2, it looks like we’re seeing a pattern here.

It would seem each number is being multiplied by 3.

So, this would mean our function is:

Now let’s check to see if this is true:

So our method works!

However, there is another way to do this:

FINDING THE RATIO

Another way to find the function we are looking for is to take the ratio.

The issue with using this way, is that you need to know for a fact that you are dealing with a geometric sequence and not an arithmetic one.

So to use the ratio method, you take one of the sequences terms

And divide it by its former term.

Like so:

So, our explicit rule is:

Now that we know how to find the geometric functions

Let’s look at a few examples:

EXAMPLE 1:

Determine the explicit and recursive rules for this sequence.

+6

+42

+294

+2058

So, looking at this we can see this is going to be a geometric sequence.

But, to really find out what we are missing let’s make sure to look at the levels to determine what the function is.

So level 1:

Well, that didn’t help much, so let’s continue:

Well, it would seem we’re multiplying by 7 each time.

Although this one is harder to see.

Either way, we can see that our explicit rule will be:

And our recursive rule will be:

That seemed unnecessarily harder.�Why not use the other way?

Well, to be fair, we used the method of finite differences on an easier problem than what we are used to.

Many times we’ll face something a little more complicated.

Like this one:

Example 2:

Determine the explicit and recursive rules for this sequence.

+6

+42

+294

+2058

So, looking at this we can see this is going to be a geometric sequence.

But, to really find out what we are missing let’s make sure to look at the levels to determine what the function is.

So level 1:

Well, that didn’t help much, so let’s continue:

Well, it would seem we’re multiplying by 7 each time.

Although this one is harder to see.

Either way, it looks like our explicit rule is:

However, let’s check to see if this actually works.

We can do that by plugging in one of the terms.

Let’s start with the zero term:

Hmmm, that didn’t work.

Well we know by using the method of finite differences that we are multiplying by 7.

But let’s see if we need to add something.

Let’s try to add c:

Now let’s find c:

So our explicit rule is:

And our recursive rule is:

SO WHAT IS A GEOMETRIC SERIES?

A geometric series is what happens when you add multiply geometric sequences together.

Essentially, you’re finding the geometric sequence.

Then adding the terms to whatever term you want to end on.

So, for example:

EXAMPLE 1:

Find the following geometric sequence to the 8^th term.

Then add the corresponding terms.

Thankfully they gave us the explicit rule for this sequence.

So to start off, let’s find our 8^th term.

So:

Now we need to find the other terms

Starting with 7

And ending with 0

Wow, that was a lot of work

Now we need to add them altogether and that will be our answer.

So:

+

Wow, that was a ton of work

And that was just to the 8^th term.

Imagine if they made us add to the 100^th term

Or the thousandth.

There has to be an easier way to do it right?

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THE EASIER WAY

As you can see, doing it the way we just did it can take a loooooooonnng time.

So, mathematicians (being the lazy people that we are) found a short cut.

To understand the shortcut though, we need to know 3 things:

• The first term in the sequence (a)
• The ratio of any two successive terms (r)
• The n power (n)

Once you have these three things, you can plug it into this formula:

So, utilizing this, we can see that from our explicit rule:

Now, we know our:

a = 9

r is also 9

and n = 8

So, plugging this in, we get:

And thankfully, we got the same answer as when we added it ourselves.

So, now let’s look at a few more examples:

Example 1:

Find the sum of the finite geometric series to the 12^th term:

So to start off, we need to find the 1^st term

So:

So our a = -5

Now we need to find the r.

Well, the r is going to be what we are multiplying by each time.

In this case, that’s -5

So r = -5

Now we need n.

They said the 12^th term

So n = 12.

Now we can plug it in!

Example 2:

Find the sum of the finite geometric series to the 7^th term:

So to start off, we need to find the 1^st term

So:

So our a = -8

Now we need to find the r.

Well, the r is going to be what we are multiplying by each time.

In this case, that’s -2

So r = -2

Now we need n.

They said the 7^th term

So n = 7.

Now we can plug it in!

So our answer is: 344

Example 3:

Find the sum of the finite geometric series to the 21^st term:

So to start off, we need to find the 1^st term

So:

So our a = -130

Now we need to find the r.

Well, the r is going to be what we are multiplying by each time.

In this case, that’s -13

So r = -13

Now we need n.

They said the 21^st term

So n = 21.

Now we can plug it in!