Now we're going to talk about sequences, both arithmetic and geometric, and I'll describe what those are. And we'll see in the future how this is all going to tie together for us. So let's start with an arithmetic sequence. So arithmetic sequence is a sequence of numbers where the pattern is that we add a constant amount to each term.
So maybe we do 3, 5, 7, 9. You know that those are the odd numbers. Okay, we can start with one if you like. These are all odd numbers, but the pattern is that we're adding two each time. Adding two to the previous term. That makes an arithmetic sequence.
We call the constant difference D, the difference between the two terms. And then we also usually call A the first term. Let's look at another example.
If our first term is 2 and our difference is 3, let's find the fifth term. So 2 is the first term, our difference is 3, so the next term is 5, 8, 11, 14. So the fifth term is 14. How about the nth term? Well, we're starting with 2, and then we do 2 plus 3, and then we do 2 plus 3 plus another 3, and then we do the previous term.
plus another 3. So in general, and this is a equals 1, 2, 3, 4. So in general we're taking the first term and we're adding the difference, let me just write 2 for a minute here, 2 plus the difference 3, how many times? This time it's 3 so that's n minus 1. or the number of terms minus one. So the number of terms minus one.
So we say a n, the nth term, is the first term plus the difference times n minus one. So in this case a n is two plus three times n minus one, where n is your term number, which term you are in the sequence. Let's find the arithmetic sequence whose fourth term is 30, and 11th term is 107. So a4 is supposed to be whatever our a is plus our difference times n minus 1 here is 3 equals 30. And then we also know a plus our difference times 10 is 107. Well, don't we have a system of equations and we can solve this.
I'll just subtract this from this. So I get my a's cancel. I get 10d's minus 3d's is 7d.
Is equal to 107 minus 30 is 77, so D is 11. So A plus 33 equals 30, so A is negative 3. Our first term is negative 3, and the difference is 11. Ultimately, this asks for the first five terms, so we have negative 3 plus 11 is 8, plus 11 is 19. First five terms, okay, plus 11 is 30, and then 41. So there's our first five terms of our sequence. Now the sum of the first n terms comes with this formula, n over 2 times the quantity 2a plus n minus 1 times d. And I believe your book develops this formula a little bit more, but I'm not going to take time to do that right now.
I'm just going to use it. Example 3, find the sum of the first 20 terms, so n equals 20, of the arithmetic sequence 3, 10, 17, and so forth. So a is 3, our difference is 7. I'm looking at the difference between two terms. So the sum of the first 20 terms is 20 divided by 2 times 2a plus n minus 1, so that's 19 times my difference 7. So I get 10 times 139, so that's 1,390 is my sum.
How about a problem with a little bit more application? A man gets a job with a salary of $30,000 a year, and he's promised a $2,300 raise each subsequent year. Find his total earnings over the first 10 years.
So we are dealing with an arithmetic sequence where our first term is $30,000. Our difference is 2,300, and we want the sum over the first 10 years. So that's supposed to be 10 over 2 times 2A, so that's 60,000. Plus n minus 1, so that's 9 times the difference, 2,300. So that's 5 times, let me calculate that a second, that's 80,700.
So our total is 403,500. Is this in the ballpark of making sense? Well, if it stayed satanic at $30,000 a year for 10 years, that would be $300,000. And we're getting a $2,300 raise each year, so this does make some legitimate sense.
that this is our total earnings for the first ten years of his job. Now let's talk about a different kind of sequence. A geometric sequence is one where instead of adding something subsequently every term, we're multiplying.
So we have A as our first term, and then we multiply by some number r, and then we multiply by r again, and then we multiply by r again. In general, our An term is A times r to the n minus 1. R is the ratio, the common ratio, of one term divided by the previous term. Now let's say in example 5, A is negative 6 and the ratio is 3. Find the first four terms. So I have negative 6 and then I'm going to multiply by 3. So I have negative 18 times 3, negative 54, negative 162. The first... four terms and the nth term is negative six times three to the n minus one.
So there's the two things we were asked for. Okay, another application problem. A ball is dropped from a height of 80 feet. The elasticity of the ball is such that it rebounds three-fourths of the distance it has fallen. So it starts...
at 80 feet. So it starts up here at 80 feet. Then it goes down and it bounces back up three quarters of that way. So if I multiply this by three quarters, that is 60. And it bounces and comes back up. So this is after one bounce.
This is after two bounces and so forth. So our common ratio is 3 quarters. We want to know how high it rebounds after the 5th bounce.
So after 1 bounce is the 2nd term. If you like, we can just keep going. Times 3 quarters gets me 45. Times 3 quarters gets me 33.75. So that's 1 bounce, 2 bounces, 3 bounces, 4 bounces. 25.31.
And after my fifth bounce, it's 18.98. How would we say that in general? We're looking for 1, 2, 3, 4, 5, 6, the sixth term, because after one bounce we get to the second term, of the sequence 80 times r to the n minus 1. So 80 times 3 quarters. to the fifth and we get 18.98 or about 19 feet.
For the next example let's look at some sequences and see if they're geometric or not. So for the first one what's the ratio? What do I multiply 2 by to get 6?
3. Do I multiply it again? Yes. Do I multiply it again? Nope that's times 2 so this is not geometric.
How about here? What do I multiply to get from e squared to e to the fourth? I multiply by e squared.
And how about the next one? e squared. The next one, e squared. This one is a yes, and the ratio is e squared. How about this one?
Multiply by 2, not multiply by 2, right? What are we doing here? This one's arithmetic, we're adding two each time. Now let's determine if it's arithmetic from its explicit formula.
So A, let's do the first four terms. So two times three to the first, so that's two times three, so that's six. And then two times three squared is 18, and then 54. 164 and then 162. So yes, this is geometric and the ratio is 3. Another way we could rewrite this is 6 times 3 to the n minus 1 and then it would look more like our normal formula, but this is still geometric. Let's look at the next one.
4 plus 3 to the n. So 3 to the first power is 3. The first one is 7. 3 squared is 9 plus 4 is 13. 3 cubed is 27 plus 4 is 31. 3 to the fourth is 81 plus 4 is 85. So is this arithmetic? Well 13 over 7, that ratio is 1.85. 31 over 13. That ratio is 2.38. So I multiply here 1.85, here 2.38.
This is not geometric. It's also not arithmetic. There's no common amount we're adding either.
Okay, now let's talk about the partial sum of a geometric. As long as r is not 1, which is a pretty unexciting sequence, the nth partial sum can be found with this formula. So let's look at our next problem.
The first term of a geometric series is sequences three. The third term is four thirds. Find the fifth term.
So let's go with our a3 equals a, which is three times r squared. And this is supposed to be four thirds. So r squared multiplied by three, I get four.
So r must be two. So as long as r is not equal to one, this is the nth sum or the nth partial sum of a geometric sequence. So let's look at this next example here.
A is 3 and the third term is 4 thirds. So we have A3, 3 times r squared is 4 thirds. So we need to divide by 3. So we have 4 over n.
So r is 2 thirds. And this could be plus or minus. I'm going to go with plus just for the sake of argument for this example.
I hadn't really thought about the fact that it could be plus or minus before now. So our fifth term, well, let's go ahead. Three times two-thirds is two. Times two-thirds is four-thirds. Times two-thirds is eight over nine.
Times two-thirds is 16 over 27. So there's our fifth term. And now we want the fifth partial sum. All right, so we want S5 is 3 times 1 minus 2 thirds to the fifth over 1 minus 2 thirds. So this is 3 times 1 minus 32 over 243 over 1 third.
So that's 243 minus 32 is 211 over 243. 43. And dividing by a third is the same as multiplying by 3. Now remember, this was actually 3 to the 5th, right? So I'm going to cancel two of those. I'm going to get 211 over 3 to the 3rd.
So 211 over 27. So that's my exact value. My approximation is 7.8 about. So both an exact answer and an approximation. So find this sum, 1 plus 3 plus 9 up to 729. So the numbers listed here are a geometric sequence. The first one is 1. The ratio is 3. So we're finding a partial sum, but we don't know how many there are yet.
So a sub n is 1 times. 3 to the n minus 1 is 729. So this is 3 to the 6th power. That's n minus 1. So n minus 1 is 6. So that must mean n is 7. So we want the 7th partial sum 1 times 1 minus 3 to the 7th. over 1 minus 3. So on top I have 1 minus 729, so that's negative 728. On the bottom I have negative 2, so 728 divided by 2 gives me a positive 364. Hmm, that's not right. I see my error now.
3 to the 7th is not 729, that was 3 to the 6th. So 3 to the 7th is 2187. So this is negative 2186 over negative 2. 2186 divided by 2 gets me 10, 1093. So I knew I had a problem because it was less than 729. So obviously it needs to be more than 729. So I get 1093 as the total sum. Okay, so now hopefully you can go between the different variations.
and notations of these different geometric and arithmetic sequences and find their partial sums.