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### Geometric Sequences

- You can use the formula for the n
^{th}term of a series to find the first term. - If you need to find a particular term and you are given the kth term, first use the kth term to find the first term. Then use the first term to find the term you are asked to find.
- When finding geometric means, remember that you might get two different values of r, one positive and the other negative. Use
*both*values to obtain two different sets of geometric means.*Each*set is a solution to the problem.

### Geometric Sequences

_{6}= − 3072 , r = 4

- Step 1 - Find a
_{1}using the formula a_{n}= a_{1}*(r)^{n − 1}and the information given - a
_{n}= a_{1}*(r)^{n − 1} - a
_{6}= a_{1}*(4)^{6 − 1} - − 3072 = a
_{1}*(4)^{5} - − 3072 = a
_{1}*1024 - a
_{1}= − 3 - Step 2 - Using a
_{1}find the explicit formula, then find the 10th term. - a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= ( − 3)*(4)^{n − 1} - a
_{10}= ( − 3)*(4)^{10 − 1} - a
_{10}= ( − 3)*(4)^{9} - a
_{10}= ( − 3)*(262144)

_{10}= − 786432

_{2}= − 8 , r = 2

- Step 1 - Find a
_{1}using the formula a_{n}= a_{1}*(r)^{n − 1}and the information given - a
_{n}= a_{1}*(r)^{n − 1} - a
_{2}= a_{1}*(2)^{2 − 1} - − 8 = a
_{1}*(2)^{1} - − 8 = a
_{1}*2 - a
_{1}= − 4 - Step 2 - Using a
_{1}find the explicit formula, then find the 10th term. - a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= ( − 4)*(2)^{n − 1} - a
_{9}= ( − 4)*(2)^{9 − 1} - a
_{9}= ( − 4)*(2)^{8} - a
_{9}= ( − 4)*(256)

_{9}= − 1024

_{6}= 243 , r = 3

- Step 1 - Find a
_{1}using the formula a_{n}= a_{1}*(r)^{n − 1}and the information given - a
_{n}= a_{1}*(r)^{n − 1} - a
_{6}= a_{1}*(3)^{6 − 1} - 243 = a
_{1}*(3)^{5} - 243 = a
_{1}*243 - a
_{1}= 1 - Step 2 - Using a
_{1}find the explicit formula, then find the 10th term. - a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= (1)*(3)^{n − 1} - a
_{12}= (1)*(3)^{12 − 1} - a
_{12}= (1)*(3)^{11} - a
_{12}= (1)*(177147)

_{12}= 177147

- Step 1) Find the common ration by dividing the second term by the first term
- r = [(a
_{2})/(a_{1})] = [12/4] = 3 - Step 2 )Write the explicit/general formula using the formula a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= a_{1}*(r)^{n − 1}

_{n}= 4*(3)

^{n − 1}

- Step 1) Find the common ration by dividing the second term by the first term
- r = [(a
_{2})/(a_{1})] = [8/4] = 2 - Step 2 )Write the explicit/general formula using the formula a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= a_{1}*(r)^{n − 1}

_{n}= 4*(2)

^{n − 1}

- Step 1) Find the common ration by dividing the second term by the first term
- r = [(a
_{2})/(a_{1})] = [12/( − 4)] = − 3 - Step 2 )Write the explicit/general formula using the formula a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= a_{1}*(r)^{n − 1}

_{n}= − 4*( − 3)

^{n − 1}

- Step 1) Find the common ration by dividing the second term by the first term
- r = [(a
_{2})/(a_{1})] = [( − 8)/( − 4)] = 2 - Step 2 )Write the explicit/general formula using the formula a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= a_{1}*(r)^{n − 1}

_{n}= − 4*(2)

^{n − 1}

- Step 1) Find the common ration by dividing the second term by the first term
- r = [(a
_{2})/(a_{1})] = [6/3] = 2 - Step 2 )Write the explicit/general formula using the formula a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= 3*(2)^{n − 1} - Step 3) Find the next three terms by evaluating a
_{5},a_{6},a_{7} - a
_{5}= - a
_{6}= - a
_{7}= - a
_{5}= 3*(2)^{5 − 1}= 3*(2)^{4}= 3*16 = 48 - a
_{6}= 3*(2)^{6 − 1}= 3*(2)^{5}= 3*32 = 96

_{7}= 3*(2)

^{7 − 1}= 3*(2)

^{6}= 3*64 = 192

- Step 1) Find the common ration by dividing the second term by the first term
- r = [(a
_{2})/(a_{1})] = [18/3] = 6 - Step 2 )Write the explicit/general formula using the formula a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= 3*(6)^{n − 1} - Step 3) Find the next three terms by evaluating a
_{5},a_{6},a_{7} - a
_{5}= - a
_{6}= - a
_{7}= - a
_{5}= 3*(6)^{5 − 1}= 3*(6)^{4}= 3*1296 = 3888 - a
_{6}= 3*(6)^{6 − 1}= 3*(6)^{5}= 3*7776 = 23328

_{7}= 3*(6)

^{7 − 1}= 3*(6)

^{6}= 3*46656 = 139968

- Step 1) Find the common ration by dividing the second term by the first term
- r = [(a
_{2})/(a_{1})] = [( − 8)/4] = − 2 - Step 2 )Write the explicit/general formula using the formula a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= a_{1}*(r)^{n − 1} - a
_{n}= 4*( − 2)^{n − 1} - Step 3) Find the next three terms by evaluating a
_{5},a_{6},a_{7} - a
_{5}= - a
_{6}= - a
_{7}= - a
_{5}= 4*(6)^{5 − 1}= 4*( − 2)^{4}= 4*16 = 64 - a
_{6}= 4*(6)^{6 − 1}= 4*( − 2)^{5}= 4*( − 32) = − 128

_{7}= 4*(6)

^{7 − 1}= 4*( − 2)

^{6}= 4*64 = 256

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Geometric Sequences

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Geometric Sequences 0:11
- Common Difference
- Common Ratio
- Example: Geometric Sequence
- nth Term of a Geometric Sequence 4:41
- Example: nth Term
- Geometric Means 6:51
- Example: Geometric Mean
- Example 1: 9th Term 12:04
- Example 2: Geometric Means 15:18
- Example 3: nth Term 18:32
- Example 4: Three Terms 20:59

### Algebra 2

### Transcription: Geometric Sequences

*Welcome to Educator.com.*0000

*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 sequence*0067

*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 a _{4} 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 a_{4} 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 term*0166

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

*So, this is a _{2}; the first term is a_{1}; therefore, if I take a_{2}/a_{2 - 1}, that is just a_{2}/a_{1}.*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, a _{6}.*0302

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

*a _{6} 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 3 ^{5}.*0354

*So, recalling powers of 3: 3 times 3 is 9, times 3 is 27; so 3 ^{3} is 27; 3^{4} is 81; and 3^{5} is 243, 81 times 3.*0359

*So, a _{6} = 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 a _{6} 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 r ^{4}; so if I take 8/2048, equals r^{4}, this simplifies to 1/256 = r^{4}.*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 4 ^{2} 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 a _{5}, 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 a _{9}(which is what I am looking for) equals*0760

*the first term, times 2 ^{8}; I am stuck.*0776

*I can't go any farther; I need the first term--I need a _{1}.*0780

*But there is something else I haven't used yet, and that is the fact that I know a _{5}.*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 a _{5} is; so let's look at this formula again and use it to solve for the first term.*0800

*a _{5} 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 r ^{4}; 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, 2 ^{4} 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 a _{1} is 5; since a_{1} = 5, let's finish this out.*0852

*a _{9} = 5 times 2^{8}; when you continue on with powers of 2, we know that 2^{4} is 16.*0861

*If you continue on up, you are going to find that 2 ^{8} is 256.*0870

*So, a _{9} = 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 out*0886

*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 a _{5} 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 = 4r ^{4}; divide both sides by 4--that gives me 81 = r^{4}.*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: a _{6} (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

1 answer

Last reply by: Dr Carleen Eaton

Sun Oct 21, 2012 7:55 PM

Post by Joe Snyder on October 15, 2012

I need help with finding the sum of a geo sequence