Algebra 2 Arithmetic Sequences

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Today, we are going to start a series of lectures on sequences and series, starting with arithmetic sequences.0002

A sequence is a list of numbers given in a certain number; and each of these numbers is called a term.0010

So, the general form that a sequence is written in is this: a₁,0016

or just a, is the first term; a₂ is the second term, and on and on.0021

Each term is given by a; and then n is the term number.0026

For example, a typical sequence would be something like 5, 10, 15, 20; and this type of sequence is called a finite sequence.0032

Another type of sequence is one like this: 3, 9, 27, 81...and then it has the three dots, the ellipses, at the end,0052

indicating that it goes on forever: this is called an infinite sequence.0064

Looking back at this second function, looking a little bit more deeply, we could actually rewrite this second function as a_n = 3ⁿ.0074

I will go into more detail about this in a few minutes.0089

But just in general, sequences in general (not just limiting ourselves to arithmetic sequences)...0092

The idea, then, would be: if I was looking for the first term, a₁ (here n = 1--it is the first term),0099

I would then substitute in 1 where the n was; so that gives me 3.0106

So, the first term is 3: a₂ is 3 to the second power, 9; a₃ is 3 to the third power, which is 27; and so on.0112

So, you can develop an equation for this sequence that will tell you what a particular term is--what the value of that term is.0126

If I wanted to know the 17^th term, then it would simply be 3¹⁷, whatever that comes out to.0134

So, this is just sequences in general; but today we are focusing on arithmetic sequences.0142

In an arithmetic sequence, each term after the first one is obtained by adding a constant--0149

not multiplying by a number or anything--it is just by adding; that is limiting this to arithmetic sequences.0154

The constant that you add to obtain the subsequent term is called the common difference.0162

In the previous slide, I showed you a typical sequence: 5, 10, 15, 12 and this is an arithmetic sequence.0171

The common difference, which is often just called d, is 5; so this is the common difference.0181

Therefore, I can start with the first term; and to get the second term, I am going to add 5 to that.0191

So, a₂ = 10; a₃, 10 + 5, equals 15; and so on.0199

One thing to be aware of is that the common difference can be negative; it also can be a fraction.0211

So, the common difference can be a positive whole number; it could be a fraction; or it could be an integer that is negative.0218

And we are going to see examples of all of those in a few minutes.0227

If you want to find the value of a particular term in an arithmetic sequence, you can use the formula for the n^th term.0233

Here, this is called the formula for the general term sometimes (either for the n^th term or the general term).0242

a_n = the first term, plus (n - 1) times the common difference.0249

So, if I know the first term and the common difference, and I am looking for a particular term (let's say the 20^th term),0257

then I would know n, so I can find the value of a₂₀.0266

Looking at a little bit different sequence here: 5, 3, 1, -1, -3, and continuing on--looking at this,0273

I can see that the numbers are getting smaller.0286

Since the numbers are getting smaller, I know that the common difference is negative.0289

And I can easily find that common difference by taking one of the terms and subtracting the previous term.0292

So, I will go ahead and take 3; I will subtract the previous term; and that is going to be equal to -2, so the common difference is -2.0299

All right, so if I have this common difference, then if I wanted to look for a particular term, I could find it.0312

For example, I may want to find a₁₀: then I could use this formula.0321

a₁₀ would equal the first term, 5, plus 10 minus 1, times the common difference, which is -2.0336

So, a₁₀ = 5 + (10 - 1)...is 9...times -2 (that is times -2, actually, making that clear).0349

a₁₀ = 5 + -18, or a₁₀ = -13.0365

Therefore, by having this formula, I can find any term in here, without going through the cumbersome of just subtracting 2, subtracting 2, and so on.0376

I knew the first term; I knew n; and I knew the common difference; the rest is just calculating.0386

OK, the equation for the n^th term: you can use the formula that I just gave in order to find the equation for the n^th term.0397

And I gave an example of applying that general equation.0404

Let's talk about using that formula to find a specific equation again.0409

So, looking at another example: 9, 12, 15, and so on--I have been given some terms in the sequence.0416

And I think back to my formula: a_n = a₁ + (n - 1)d.0431

I want to find an equation for the n^th term specific to this sequence.0441

So, I am going to look at what I have: I have a₁...I have the first term, often just called a; that equals 9.0446

I need to find the common difference: I can find the common difference by saying 12 - 9 = 3.0453

Therefore, I can write a general equation: the general term, or the n^th term, a_n, equals 90459

(9 is the first term), plus (n - 1), times the common difference, which is 3.0472

So, a_n = 9 + 3n - 3; a_n = 6 + 3n.0479

Now, I have a formula that I can use; so if I wanted to find any term, such as a₂₁, I could use this.0493

a₂₁ = 6 + 3(21); so, a₂₁ would then equal 63 + 6, which equals 69.0504

OK, so again, I have my general equation here; and then I said, "All right, I have my general formula; I can write a specific equation for this situation."0532

The first term is 9; and the common difference is 3; that comes out to 9 + 3n - 3; 9 - 3 is 6, so it is 6 + 3n.0544

Now, if I want to find the 21^st term, then I will say, "OK, I am going to go ahead and use my specific equation0558

for this scenario, for this sequence," which is a₂₁ = 6 + 3(21).0570

That gives me 63 + 6; a₂₁ is 69.0582

Arithmetic means: arithmetic means does not mean finding the mean of the sequence or anything; this is totally different.0588

Arithmetic means are actually the terms between two non-successive terms of an arithmetic sequence.0595

And you can actually use the formula that we learned for the n^th term to find the common difference, d.0603

Once you know the common difference, d, you can use d to find the arithmetic means between terms.0610

For example, if I were given a sequence, -3...and then there were some missing terms: one term is missing,0616

two terms are missing, three, four, and then they give me the last one; I have my first term, a₁;0625

and I have 1, 2, 3, 4, 5, 6 terms in this; so a₁ = -3, a₆ = 17.0634

Then, I look back to that formula for the general term, a_n = the first term, plus (n - 1) times the common difference.0645

In this case, I am going to be using the formula, not to find a_n yet; first I need to find the common difference.0654

I then use the common difference to find these terms.0661

Let's go ahead and find the common difference.0664

If a₆ is 17, and I put that here; -3 is the first term, and I am working with a₆;0666

so in this case, there are 6 terms, a₆...6 - 1 times the common difference, which is what I am looking for.0675

Now, it is just a matter of solving for that: this gives me 17 = -3 + 5d; add 3 to both sides; that is going to give me 20 = 5d.0685

And then, I am going to divide 20 by 5 to give me 4 = the common difference.0699

So now that I have a common difference of 4, all I have to do is find a₂; that is going to equal -3 + 4, so that is going to equal 1.0706

To find a₃, it is going to be 1 + 4 = 5; a₄ = 5 + 4; that is 9.0715

And a₅ = 9 + 4, which is 13.0726

OK, we have -3; and now, our missing terms: 1, 5, 9, and 13; and then I have 17, which was already given.0732

So, I found the arithmetic means, or the missing terms, by using the formula for the n^th term0746

to find the common difference, and then taking that common difference and adding it to each term to find the next term.0750

Example 1: Write an equation for the n^th term of this sequence.0758

Recall that the equation for the n^th term, just the general formula, is a_n, the general term,0764

equals the first term, plus (n - 1) times the common difference.0774

Therefore, we need to find the common difference; and you can find that by taking any term and subtracting the one just before it.0779

Therefore, 26 - 19 = 7; so the common difference is 7.0786

a_n equals the first term; well, I also have the first term--that is -2.0797

So, that is -2 + (n - 1) times 7, so a_n = -2 + 7n - 7; a_n =...-2 + -7 is -9, so 7n - 9.0801

So, this is the equation to find any term of this sequence.0822

Example 2: Find the arithmetic means: this time, we need to find the missing terms.0829

And we can do that because we, again, have that equation for the n^th term,0834

which is a_n = the first term + (n - 1) times the common difference.0840

In order to do this, though, I need to find the common difference; and I can do that because I have the first term,0846

which is -7; I also have 1, 2, 3, 4, 5, 6, 7...I also have the seventh term, which is equal to 11.0854

And in this case, n = 7; so I just go ahead and use these values to find d.0863

Therefore, I am going to end up with 11 = the first term, which is -7, plus n, which is 7, minus 1, times d.0872

Therefore, 11 = -7 + 6d; I am going to add 7 to both sides to simplify--that gives me 18 = 6d.0887

Divide both sides by 6; I end up with d = 3.0896

Now that I have this, all I need to do is say, "OK, for the second term" (I have the first term),0900

"I am going to take the first term, -7, and just add 3 to that to get -4."0908

For a₃, I am going to take -4 and add 3 to that to get -1.0914

a₄ = -1 + 3, so that is going to give me 2.0925

a₅ = 2 + 3, which is actually equal to 5; and then, a₆, which is also missing, is going to be 5 + 3, or 8.0933

Therefore, the term I was given was -7; and then, I found the missing terms, -4, -1, 2, 5, and 8; these were the missing terms, the arithmetic means.0943

And I was given the last term, 11.0960

And the key thing is to find the common difference, using this general equation, and then to take that common difference and use it to fill in the missing terms.0962

Which term is 763 in the arithmetic sequence (and the arithmetic sequence is given)?0974

So, you have a term, a_n = 763; and what this is really asking is what place that is.0983

Is it the fifth term? Is it the seventeenth term? What number term is this?0991

I know that its value is 763; and let's look at what else I know.0997

Well, I know that the first term is -7; and I can easily find the common difference.1002

I can just take 15 - 4, for example, which is 11; so knowing this value, the first term, and the common difference,1009

I can go back to my equation for the n^th term, a_n1017

equals the first term, plus n - 1 times the common difference.1022

What I am looking for is n, the term number: for a_n = 763, what is n?1027

I can put in 763 = -7 +...n is my unknown, so I have (n - 1) times this common difference of 11.1037

So, 763 = -7 + 11n - 11; adding 7 to both sides is going to give me 770 = 11n - 1.1047

Now, adding 11 to both sides gives me 781 = 11n; and the last step is just to divide both sides by 11.1065

And if you figure that out, it comes out to n = 71.1074

Therefore, the term number equals 71, or a₇₁ = 763.1078

So, this time, I was given a term, and was asked to figure out which term it is in the sequence.1090

Where does it land in this sequence?--I could do that because I had the first term; I had the common difference; and I had the value of that term.1097

All right, find an equation for the n^th term of the arithmetic sequence with a₁₀₁ = 100 and a common difference of 7.1106

Well, to figure out this equation for the n^th term, I need the first term.1118

So, I have the common difference; but the thing that is missing here is the first term.1125

So, how do I figure that out? Well, I do know another term; I know an a_n.1138

Since I know a_n, or a₁₀₁, and I know that n is 101 in that case, and I know the common difference, I can solve.1143

So, I am going to first solve for the first term.1151

Then I will go back and use that first term to develop an equation for the n^th term for this sequence.1156

So, I have a₁ = 100, and I don't know my first term; I know my n for this term is 101,1163

and I am going to say minus 1, times the common difference, which is 7: this gives me 100 = a₁ + 100 times 7.1176

100 = a₁ + 700; subtract 700 from both sides; that is going to give me...actually, let's see:1187

yes, it is going to give me -600 equals this first term.1202

So, I am just rewriting it in a more standard form: the first term equals -600.1207

Now that I know the first term, I go back again and look at that general equation for the n^th term.1212

And recall that I am asked to find a specific equation for the n^th term for this sequence.1220

And I can do that, because I know that the first term is -600, and the common difference, d, is equal to 7.1226

So, a_n = -600 + 7n - 7; this equals a_n = 7n - 607, just simplifying.1236

I had to take an extra step here, because I wasn't given the first term.1251

But since I was given another term, I could find the first term.1254

And then, I went ahead, and I used that to find the equation for the n^th term for this sequence, which is a_n = 7n - 607.1258

This concludes the lesson on arithmetic sequences on Educator.com; thanks for visiting!1269