Algebra 2 Geometric Sequences

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In the previous lessons, we talked about arithmetic sequences and series.0002

So, we are going to go on to discuss geometric sequences.0007

What are geometric sequences? Recall that a sequence in general is a list of numbers in a certain order.0012

So, it is in the general form first term, second term, and on...and it could end at a particular term, a_n, or it may go on indefinitely.0020

In previous lessons, we talked about arithmetic sequences: for example, 5, 10, 15, 20.0034

And each term was related to the previous one by a common difference, d (here d = 5).0043

What we did is added whatever the common difference was to a term to get the next, to get the next, and so on.0051

So, if you need to review arithmetic sequences or sequences in general, it might be a good idea to go back and start with that lecture.0057

And now, we are going to continue on and learn about geometric sequences.0064

So again, a geometric sequence is a list of numbers; so a geometric sequence is a sequence0067

in which each term after the first is found by multiply the previous term by a non-zero constant r.0073

When we talked about arithmetic sequences, we added the common difference.0081

Here we are going to multiply a term by the common ratio r to get the next term.0087

Therefore, a_n equals the previous term, which is a_{n - 1}, times the common ratio.0092

So, if I were to look for a₄ in a particular series, I would find it by saying,0103

"OK, the previous term, a_{4 - 1}, times the common ratio..." so the term a₄ would equal r times the third term.0109

Looking at an example of this, this, again, was an arithmetic sequence.0124

Now, we are going on and talking about an example that gives you a geometric sequence: 3, 12, 48, 192.0150

Working with these, it is important to find a common ratio: the common ratio is given by taking a term0166

(any term--I am going to take 12) and dividing it by the previous term.0173

So, this is a₂; the first term is a₁; therefore, if I take a₂/a_{2 - 1}, that is just a₂/a₁.0178

So, take a term; divide it by the previous term; and this is going to give me the common ratio of 4.0191

So, you can see where these two equations come from.0206

To find the next term, you multiply the term previous to it by the common ratio.0209

To find the common ratio, you take a term, and you divide it by the term that came just before.0215

Now, as we talked about arithmetic sequences, I said that the common difference could be a negative number; it could also be a fraction.0223

And that is true, as well, with the geometric series.0231

For example, I could have 2, -6, 18, -54: this is another geometric sequence, and I want to find the common ratio.0237

So, I will take a term; I am going to go for -6, and I am going to divide it by the previous term, which is 2; and it is going to give me -3.0250

The thing that you will notice is: when you have a negative common ratio, the terms are going to alternate their signs.0257

So, I am going to have a positive, a negative, a positive, a negative.0265

If it is a positive common ratio, then the terms will just all be positive.0269

You can also have a common ratio that is a fraction; and we will work with those examples later on in the lesson.0274

For a geometric sequence, there is a formula for the n^th term.0282

So here, the formula for the n^th term is that the general term, a_n,0286

equals the first term, times the common ratio, taken to the power n - 1.0290

Looking at a geometric sequence: 2, 6, 18, and we will make this an infinite sequence...0296

it is going to go on and on...perhaps I was asked to find the sixth term, a₆.0302

What is that? Well, I can use this formula.0309

a₆ equals the first term (I have the first term--it is 2), and I also need to find the common ratio.0312

The common ratio is going to be any of the terms (I will take 18), divided by the previous term (6), which is 3.0320

So, I have the common ratio; I have the first term; and I have n; here, since I am looking for the sixth term, n will equal 6.0329

So, the sixth term is equal to the first term (which is 2), time the common ratio (which is 3), raised to the power of 6 - 1.0339

The sixth term is equal to 2, times 3⁵.0354

So, recalling powers of 3: 3 times 3 is 9, times 3 is 27; so 3³ is 27; 3⁴ is 81; and 3⁵ is 243, 81 times 3.0359

So, a₆ = 2 times 243, or simply 486.0381

So again, this is the formula for the n^th term of a geometric sequence.0389

Given the sequence, I could find the sixth term, because I know the first term, 2; I know the common ratio;0394

I was able to figure out that it is 3; and I know that n is equal to 6; so I got that a₆ is 486.0404

Geometric means: again, thinking back to arithmetic sequences, we said that arithmetic means are missing terms in an arithmetic series.0412

And that is analogous to this situation: geometric means are missing terms between two non-successive terms of a geometric sequence.0421

2048 is the first term; then I have three missing terms--those are the geometric means; and then, I have my last term, 8.0429

Use the common ratio to find the geometric means.0442

If I find the common ratio, then all I need to do is take a term and multiply it by the common ratio to find the next term;0444

multiply that by the common ratio to find the next term; and so on.0450

If I want to find the missing terms here, I am going to use my formula for the n^th term,0454

a_n = the first term, times r^{n - 1}.0458

Let's look at what I am given: the first term is 2048: 1, 2, 3, 4, 5...the fifth term is 8; OK.0463

Using this formula: a...I need to find the common ratio, and I can do that because I have the first and last terms, and I have n.0476

1, 2, 3, 4, 5; n = 5, right here; so a_n is 8; that is equal to the first term, times the common ratio, raised to the power 5 - 1.0485

8 = 2048 times r⁴; so if I take 8/2048, equals r⁴, this simplifies to 1/256 = r⁴.0501

If you think about your roots and your powers, the fourth power is actually plus or minus...0524

I am going to take the fourth power of 1/256; it is actually ± 1/4.0535

And if you multiply this out, you will find that 4² is 16, times 2 is 32, times 2 is 64...0540

Excuse me, 1/4 times 1/4 is 1/16, and continuing on, you will find that the fourth power of 1/4 is 1/256.0551

OK, the important thing to note, though, is that it is not just that the fourth power is 1/256; it is actually plus or minus,0568

because I could take -1/4 times -1/4 times -1/4 times -1/4, and I would also get 1/256.0579

Therefore, with geometric means, you may end up with two sets of answers.0591

All right, so I found my common ratios, which could be ± 1/4; and I have my first term, 2048.0599

So, to find my second term, I just take 2048 times 1/4; that is 5/12.0606

To find my third term, I just take 512, times 1/4; and I am going to get 128.0614

To find the fourth term, I am going to take 128 times 1/4, and I am going to get 32.0626

So, that is one set of answers; I actually have two sets of answers.0635

If r = 1/4, then the missing terms (the geometric means) are 512, 128, and 32.0639

If r is actually equal to -1/4, then the signs will alternate, so what I am going to get is 2048, then -512, then positive 128, then -32.0665

So, I have two possible sets of geometric means.0682

Again, to find the geometric means, I am just going to find the common ratio.0686

And I was able to do that using this formula; I got two answers--the common ratio is either plus or minus 1/4.0691

I took the first term; I multiplied it by 1/4, and then multiplied that by 1/4, and on to get this set of geometric means.0698

I took my other possible solution, r = -1/4, and multiplied 2048 by that to get -512, times -1/4 is 128, times -1/4 is -32.0705

So, that is something to keep in mind: that, with geometric means, you can get two sets of solutions.0718

Example 1: Find the ninth term of the geometric sequence with a fifth term of 80 and a common ratio of 2.0726

The formula for the n^th term is first term, times r^{n - 1}, the common ratio to the n - 1 power.0734

Here, n = 9; and I am given a₅, and I am already given r.0744

So, let's go ahead and work on this.0755

I have my common ratio of 2, and n is 9, so that is 9 - 1; so a₉(which is what I am looking for) equals0760

the first term, times 2⁸; I am stuck.0776

I can't go any farther; I need the first term--I need a₁.0780

But there is something else I haven't used yet, and that is the fact that I know a₅.0786

In order to go over here and find the first term, I can do that by saying, "OK, let's look at this formula again."0791

I know what a₅ is; so let's look at this formula again and use it to solve for the first term.0800

a₅ equals the first term, times r; in this case, n will be 5; and that is to the 5 - 1.0809

So, I can substitute in; I know that 80 equals the first term, times r⁴; and I already do know r, so let's put that in, as well.0818

The first term is 2 to the fourth; 2 times 2 is 4, times 2 is 8, times 2 is 16.0827

So, 2⁴ is actually 16; if I divide both sides by 16, I am going to get 80/16, and the first term is 5.0838

Now, I can go back and finish my problem.0849

I know that a₁ is 5; since a₁ = 5, let's finish this out.0852

a₉ = 5 times 2⁸; when you continue on with powers of 2, we know that 2⁴ is 16.0861

If you continue on up, you are going to find that 2⁸ is 256.0870

So, a₉ = 5(256); 5 times 256 is 1280, and that is what we were looking for.0875

All right, again, it looked straightforward, but we had to take a detour, because when we started out0886

using that formula to find the ninth term, we discovered that we got stuck at this step, because we didn't have the first term.0893

But, since they gave us another term and the common ratio, I was able to go back,0902

substitute in 80 here, put in my common ratio of 2, and solve for the first term.0906

Then, I finish out the problem to find that the ninth term is equal to 1280.0914

Find the geometric means: so we need to find the three missing terms.0919

I always look at what I am given first.0923

Well, I am given the first term, and I am given 1, 2, 3, 4, 5...the fifth term.0925

As always, I am going to use my formula here, that the n^th term is equal to the first term, times the common ratio raised to the n - 1 power.0934

To find the geometric means, I need the common ratio--what is r?0944

If I have r, I multiply it by 4 to get the second term, then the second term by the common ratio to get the third, and so on.0948

But I don't know r: what I do know are these two things, so I can find r.0957

I have that a₅ is 324; and my first term is 4; in this case, n is going to be 5; now I can find the common ratio.0961

This gives me 324 = 4r⁴; divide both sides by 4--that gives me 81 = r⁴.0978

The fourth root of 81 is 3; 3 to the fourth power is 81.0988

But there is something I have to remember: the other fourth root of 81 is -3.0995

If I take -3 times -3 times -3 times -3, that is 9, -27, times -3 is also 81.1002

So, I have two possibilities here: r can equal plus or minus 3.1013

I am going to have two sets of results here.1023

Let's let r equal 3; if r equals 3, then I am going to end up with 4 as my first term; I add 3 to that--I am going to get 7.1027

I am going to go ahead and (let's see) add 3 to that...1039

Actually, a correction: I was thinking of arithmetic series; this is a geometric series--I am going to multiply.1051

I need to multiply each term, so 4 times 3 is going to give me 12, times 3 is going to give me 36, times 3 will give me 108, times 3 is 324.1057

So, make sure, when you are working with geometric series, that you are multiplying, not adding.1076

So again, if r = 3, I am going to get geometric means of 12, 36, and 108.1080

If r equals -3, I am going to get 4 times -3 is -12; -12 times -3 is going to give me positive 36, times -3 is -108.1087

They are alternating signs; so there are two possible solutions for the geometric means: 12, 36, 108; or -12, 36, -108.1100

Write an equation for the n^th term of the geometric sequence -2, 1/2, and -1/8.1112

The formula for the general term is the first term, times the common ratio raised to the n - 1 power.1123

So, if I am looking for an equation for the n^th term here, I am going to need the common ratio.1133

To find the common ratio, I will just take a term and divide it by the previous term.1140

The common ratio...I could take 1/2, and I am going to divide that by -2.1145

Recall that I could just rewrite this, to make it a little clearer, as 1/2 divided by -2--just write it out.1148

And that is the same as multiplying 1/2 by the inverse of -2, and the inverse of -2 is -1/2.1156

So, that is -1/4; the common ratio is -1/4.1171

Now, I can go ahead and write my equation: a_n = the first term, which is -2, times (-1/4)^{n - 1}.1178

And I only have three terms here; but just writing it in a more general form...n - 1, because it is just asking me for the n^th term.1194

-2 times -1/4...a negative times a negative is going to be a positive, so that is just going to be 2/4, or 1/2.1210

Oh, actually, I cannot simplify that any further--correction.1225

I can't simplify that any further, because it is (-1/4)^{n - 1}; we are actually done at this step.1228

We are done right here with the general formula, because I don't have n.1234

All right, so the equation for the n^th term is simply going to be a_n = -2(-1/4)^{n - 1}.1238

So, I could find any term from this geometric sequence, using this equation.1252

All right, write the next three terms of the geometric sequence: -1/3, 1/2, -3/4.1260

In order to find a term, I need to have the common ratio.1268

So, let's find that common ratio by taking 1/2 and dividing it by the previous term, which is -1/3.1272

This is the same as 1/2 divided by -1/3; and remember, I can always rewrite that as 1/2 times the inverse, which is -3, or -3/2.1279

Therefore, r = -3/2; now that I have the common ratio, I can find the next three terms.1290

So, we stopped with the third term--I am looking for the fourth term, the fifth, and the sixth.1297

So, the fourth term is going to be equal to -3/4, times -3/2; this is just going to be 9/8.1306

The fifth term is going to be equal to 9/8, times that common ratio of -3/2.1317

-3/2 times 9/8 is going to give me -27/16: a₆ (the sixth term) is going to be -27/16 times -3/2.1326

A negative and a negative is going to give me a positive, and 27 times 3 is actually 81.1343

16 times 2 is 32; and you could leave these as fractions, or you could rewrite them as mixed numbers.1349

I am just going to leave them as they are; so the next three terms are 9/8, -27/16, and 81/32.1357

And I could have looked here and just predicted that the common ratio is negative, because I have these alternating signs: negative, positive, negative.1366

That concludes this lesson on Educator.com, covering geometric sequences; thanks for visiting!1376