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Geometric Sequences & Series

  • A sequence is geometric if every term in the sequence can be given by multiplying the previous term by some constant number r:
    an = r·an−1.
    We call r the common ratio (since r = [(an)/(an−1)]). Every "step" in the sequence multiplies by the same number. The number can be anything, so long as it is always the same for each step.
  • The formula for the nth term (general term) of a geometric sequence is
    an = rn−1 ·a1.
  • To find the formula for the general term of a geometric sequence, we only need to figure out its first term (a1) and the common ratio (r).
  • We can use the following formula to calculate the value of a geometric series. Given any geometric sequence a1, a2, a3, …, the sum of the first n terms (the nth partial sum) is
    Sn = a1 · 1−rn


  • We can find the partial sum (Sn) by only knowing the first term (a1), the common ratio (r), and how many terms are being added together (n). [Caution: Be careful to pay attention to how many terms there are in the series. It can be easy to get the value of n confused and accidentally think it is 1 higher or 1 lower than it really is.]
  • Unlike an arithmetic series, we can also consider an infinite series when working with a geometric sequence. If |r| < 1, then
    S   =  a1 · 1



Geometric Sequences & Series

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
  • Introduction 0:06
  • Definition 0:48
  • Form for the nth Term 2:42
  • Formula for Geometric Series 5:16
  • Infinite Geometric Series 11:48
    • Diverges
    • Converges
  • Formula for Infinite Geometric Series 16:32
  • Example 1 20:32
  • Example 2 22:02
  • Example 3 26:00
  • Example 4 30:48
  • Example 5 34:28

Transcription: Geometric Sequences & Series

Hi--welcome back to

Today, we are going to talk about geometric sequences and series.0002

The other specific kind of sequence we will look at in this course is the geometric sequence,0006

a sequence where we multiply by a constant number for each step.0011

In the previous lesson, we looked at the arithmetic sequence, which is where you add by a constant number.0013

Now, we are looking at geometric, where you multiply by a constant number every step.0018

Just like arithmetic sequences, geometric sequences commonly appear in real life.0022

Since geometric sequences are based on ratios, since we are always multiplying by the same thing,0027

and ratios occur a lot in the world, they give us a way to describe a wide variety of things.0030

In this lesson, we will begin by going over what a geometric sequence is, and how we can talk about them in general.0036

Then, we will look into formulas for geometric series to make adding up a bunch of terms really easy and fast; let's go!0041

We will start with a definition: a sequence is geometric if every term in the sequence can be given0047

by multiplying the previous term by some constant number, r.0053

an is equal to r times an - 1; that is, some term is equal to the previous term, multiplied by r.0058

We call r the common ratio, because we can express it as an divided by an - 1; that is, some term divided by the previous term.0073

And so, we have a ratio in the way that we are building it.0084

Here are two examples of geometric sequences: 3, 6, 12, 24...continuing on; 4/5, -4/25, 4/125, -4/625...continuing on.0087

They are geometric, because each step multiplies by the same number.0099

For example, in this one, every step we go forward, we are multiplying by 2: 3 to 6--times 2; 6 to 12--times 2; 12 to 24--times 2.0103

And this is going to continue on forever, as long as we keep going with that sequence.0112

Over here, 4/5 and -4/ is not quite as simple, but it is basically the same thing, multiplying by -1/5.0116

That is how we get from 4/5 to -4/25, if we are going to multiply.0124

To get to 4/125, once again, we multiply by -1/5; to get from 4/125 to -4/625, we multiply by -1/5.0128

And this is going to keep going, every time we keep stepping.0139

Every step is multiplying by the same number: we are multiplying by the same number each step.0143

The number can be anything: it can positive; it can be larger than one; it can be less than one; it can be negative.0148

It doesn't matter, so long as it is always the same value for every step.0153

The definition of a geometric sequence is based on the recursive relation an = r(an - 1).0159

That is, every term is equal to the previous term, multiplied by r.0165

How can we turn this into a formula for the general term, where we don't have to know what the previous term is--0170

we can just say, "I want to know the nth term," plug it into a formula, and out will come the nth term?0173

Remember: a recursive relation needs an initial term.0179

So, while this relation that defines a geometric sequence is useful, we still need a little bit more.0181

We need this initial term to know where we start--what our very first term is--because previous to that...there is nothing previous.0187

So, we just have to state that as one specific term.0193

Since we don't know its value yet, we will just leave it as a1, our first term.0197

From an = r(an - 1), we see that a1 relates to later terms as: a2 will be equal to r(a1).0202

The second term will be equal to the first term, times r; this will continue on.0210

The third term, a3, will be equal to r times a2.0215

But we just showed that a2 is equal to r(a1), so we can plug that in there;0218

and we will have r times r(a1), so we end up getting r2(a1) = a3.0224

We can continue on here; a4 is going to be equal to r times a3.0233

But we just figured out that a3 is equal to r2 times a1.0237

So, we replace a3, and we end up having r times r2 times a1;0241

so now we have that r3(a1) = a4.0246

And we will see that this pattern will just keep going like this.0251

So, we have that a1 is equal to a1 (there is no big surprise there).0254

a2 is equal to r times a1; a3 is equal to r2 times a1.0258

a4 is equal to r3 times a1; and we see that the pattern is just going to keep going like this.0263

The nth term is n - 1 steps away from a1; it is n - 1 steps to get from a1 to n.0268

If you start on the first stepping-stone, and you go to the nth stepping-stone, you have to take n - 1 steps forward.0277

We start at a1; to get to the an, we have to go n - 1 steps forward.0283

Since every step means multiplying by r, that means we have multiplied by r n - 1 times, which is r raised to the n - 1 power.0287

Thus, we have that an = rn - 1(a1).0296

So, to find the formula for the general term of a geometric sequence, we only need to figure out what the first term is, and the common ratio.0301

As soon as we figure out a1 and our value for r, we have figured out what the general term is, what the an term is--pretty great.0308

What if we want to find the nth partial sum of a geometric sequence (that is, adding up0316

the first n terms of the sequence, a1 + a2 + a3 up until + an)?0320

Well, we could just add it all up by hand for small values of n.0327

If it was n = 2, so it was just a1 and then a2, it is probably not that hard to just figure it out by hand.0331

If it was n = 3, we could probably do it by hand; if it was n = 10, n = 100, n = 1000,0336

this gets really, really tiresome, really quickly, as the value of n gets larger and larger.0341

So, how can we create a formula--how can we just have some formula where we can plug some stuff in,0345

and we will immediately know what that nth partial sum is?0350

Well, let's do two things: first, let's give the sum a name.0355

So, we will call our nth partial sum sn, the sum for the nth partial sum.0358

Second, let's use the form for the nth term of a geometric sequence.0364

Remember: we just figured out that the general form for an is rn - 1(a1).0367

We can use this general term to put of the terms in this series into a format that will involve a1.0376

So, we have sn = a1 + a2 (which is r times a1) + a30382

(which is now r2 times a1), up until we get to + an (which is now rn - 1(a1)).0390

Great; but at the moment, we can't do anything with just this.0399

This isn't quite enough information; we can't combine the various r's, because they all have different exponents.0402

r to the 0, what is effectively here; r1; r2; r3; up until rn - 1--0410

they don't talk the same language, because they don't have the same exponent.0417

So, since they can't really communicate with each other, we can't pull them out all at once.0420

We could pull out all of the a1's, but then we would still be left with all of these different kinds of r's.0423

So, we don't really have a good thing that we can do right now.0428

What we want, what we are really looking for, is a way to somehow get rid of having so many things to add up.0431

We want fewer things to add up; if we only had a few things to add up and compute, it would be easy for us to calculate these values.0439

So, that is what we want to figure out how to do.0445

This is the really clever part; this is basically the part of the magic trick where suddenly we pull the rabbit out of the hat.0449

And so, you might see this and think, "How would I figure this out?"0455

And it is a little bit confusing at first; but just like, as you study magic more and more (if you were to study magic),0458

you would eventually realize, "Oh, that is how they got the rabbit into the hat," or "this is how the trick works"--0463

as you work with math more and more, you will be able to see, "Oh, that is how we can make these sorts of things."0468

So, don't be worried by the fact that you would not be able to think of this immediately.0472

The people who made this proof at first didn't think of it immediately.0477

They thought about it for a while; they figured out different things, and maybe they tried something that did not work;0481

and eventually they stumbled on something and said, "Oh, if I do this, it works,"0486

and they were able to come up with this really easy, cool, clever way to do it.0489

But it is not something that you just have immediately.0492

It is something that you have to think about, until eventually you can "pull your own rabbit out of a hat."0494

But first, you have to get the rabbit into the hat.0498

Anyway, here is the clever part: what is it?0500

We have sn = a1 + r(a1) + r2(a1) + ... + rn - 1 (a1).0503

The really cool trick that we do is say, "What if we multiplied r times sn?"0515

Well, that would end up distributing to everything in here, since we are multiplying it on both sides of the equation.0522

We would have r(a1) + r2(a1) + r3(a1)...0528

all the way until we get up to + rn(a1).0534

Now, notice: we now have matching stuff going on.0537

Here is r times a1; here is r times a1; there is a connection here.0540

Here is r2 times a1; here is r2 times a1; there is a connection there.0544

Here is r3 times a1; and somewhere in the next spot in the decimals is r3 times a1, as well.0548

Here is rn - 1 times a1; and in the next back spot in the decimals, here is rn - 1 times a1.0555

We have all of this matching going on; well, with this idea, since we have all of this matching,0560

we can subtract this equation, the rsn equation, this one right here,0565

from the sn equation by using elimination.0569

From when we talked about systems of linear equations: if we have two equations, we can subtract, and we can add them together.0572

We are just using elimination to do this.0578

So, we have our sn equation here, and then we subtract by minus r times sn.0580

So now, we have this matching pattern going on: r times a1 matches to -r times a1; they cancel each other out.0586

r2 times a1 matches to minus r2 times a1; they cancel each other out.0592

Everything in the dots here cancels out with all of the negatives in the dots here.0596

We finally get to rn - 1 times a1 in our top equation,0600

which cancels out with minus rn - 1 times a1 in our bottom equation.0603

The only things that end up being left over are minus rn times a1 and +a1 here.0608

We get sn -rsn is equal to a1 - rn(a1).0617

So, through this immense cleverness (and once again, this isn't something that you would be expected to just know immediately,0624

and be able to figure out really easily--this is the part that takes the really long thinking.0629

This is the really clever part; this is what takes hours of thought, by just thinking, "I wonder if there is a clever way to do this."0632

And eventually, you end up stumbling on it.0637

So, through immense cleverness, we have shown that sn - rsn = a1 - rn(a1).0640

That is pretty great: we have gotten this from what we just figured out.0648

Now, our original goal was to find the value of the nth partial sum, which was sn.0653

Using the above, we can now solve for sn: we just pull out the sn,0658

so we have sn times 1 minus r; we can also pull out the a1 over here on the right side.0663

We divide both sides by 1 - r, and we get that the nth partial sum is equal to0667

the first term, a1, times the fraction 1 - rn over 1 - r.0672

So, we now have a formula to find the value of any finite geometric series at all, really easily.0678

All that we need to know is the first term, a1; the common ratio, r,0683

which shows up on the top and the bottom; and how many terms are being added together total (the n exponent on our top ratio).0688

It is pretty great--a really, really powerful formula that lets us do a lot of what would be very tedious, very slow, difficult addition, just like that.0696

But what if the series was not finite?0705

So far, we have only talked about if we are doing a finite sum of the sequence.0707

But what if we had an infinite sequence, and we wanted to add up infinitely many of the terms, so we kept adding terms forever and ever and ever?0711

To understand this better, let's consider some geometric sequences and what happens as we take partial sums using more and more and more terms.0718

First, we will look at 3, 9, 27, 81, it is times 3 each time; it is a geometric sequence.0725

So, our first partial sum, s1, would be just the 3.0732

Our next partial sum would be s2, so we add on the 9, as well; we get 12.0737

Our next partial sum would be s3; we add on the 27, as well; we get 39.0741

The next partial sum is s4; we add on the 81; we get 120.0745

The next partial sum: we add on 243; we get 363; and this is just going to keep going on in this pattern.0749

It is going to keep adding more and more and more.0754

We look at this, and we notice that, as the partial sums use more and more terms, it continues to grow at this really fast rate.0756

In fact, the rate is going to get faster and faster and faster as we add more and more terms.0763

We see its rate of growth increasing as it goes to larger sums.0766

So, if we add terms to the series forever, it is not going to really get to anything.0770

It is going to blow out to infinity; it is going to just "blast off" to infinity.0774

There is no stopping this thing; it is not going to give us a single value.0778

It never stops growing; we say that such an infinite series, one that never stops growing, that doesn't go to a single value, "diverges."0781

As we add more and more terms, it continues to change forever.0789

It diverges from giving us a single, nice, clean value, because it instead just blows off to infinity.0792

It keeps moving around on us; it doesn't stay still; it doesn't go to something; it just goes off, so this would be a divergent series.0798

On the other hand, we could consider another partial sum from this below geometric sequence: 1, 1/2, 1/4, 1/8, 1/16...0807

What we are doing each time here is dividing by 2, or multiplying by 1/2; so we see that this is a geometric sequence.0815

Our first partial sum would be s1 = 1; we just add in that first term.0821

The next one, s2, would be 1.5, because we added 1/2, so we are at 1.5.0826

The next one, s3: we add in 1/4; that is 0.25 added in; that becomes 1.75.0831

The next one, s4: we add in an 8; that becomes 1.875.0836

The next one, s5: we add in 1/16; that becomes 1.9375.0840

And it would continue in this way; but we notice that it is not really growing the same way.0844

This time, the sums are continuing to grow, but the rate of growth is slowing down with each step.0849

It is not increasing out like the previous one (it was blowing out somewhere; it was becoming really, really big).0856

But with this one, we see it settling down; as we add more and more terms, it is going to a specific value.0861

It is going to 2; the infinite sum, this infinite series, is going to a very specific value; it is working its way towards 2.0868

If you keep adding more, you will see even more, as it gets to 1.99, 1.999, 1.9999999...0876

As you keep adding more and more terms, you will see that it is really just working its way to a single value.0883

It is slowing down as it gets to 2.0887

In this case, we say that such an infinite series converges; it is converging on a specific value.0889

As we add more and more terms, it works its way towards a single value.0895

There is this single value that it is working towards.0899

From the two examples we have seen, we see that whether a series converges or diverges is based on the common ratio of its underlying sequence.0902

If the common ratio is large, it causes the sequence to always grow.0911

It keeps growing, because that ratio keeps multiplying it to get larger and larger and larger, moving around.0917

Specifically, if the absolute value of r is greater than or equal to 1, the partial sums will always be changing,0922

because the size of our terms never shrinks down; so the series will diverge.0927

On the other hand, if the common ratio is small, it causes the sequence to shrink down.0933

If we have a small common ratio, it is going to make it smaller and smaller and smaller with every term we work on.0937

Since it gets smaller and smaller and smaller, we have that if the absolute value of r is less than 1,0944

the rate of change for the series will slowly disappear to nothing, because every time we go to the next term,0948

since r is less than 1, it makes it smaller; and then it makes it smaller; and then it makes it smaller, and makes it smaller,0954

and makes it smaller, and makes it smaller; so every time it is getting smaller.0959

So, every time, the rate of growth is going down to less and less and less.0962

And so, over the long term, over that infinite number of terms, it ends up converging to a single value.0965

So, as a general rule, an infinite geometric series will converge if and only if the absolute value of r is less than 1.0971

So, the absolute value of r being less than 1 means that the infinite geometric series converges.0979

If the infinite geometric series converges, then the absolute value of r must be less than 1; they are equivalent things for a geometric sequence.0983

All right: assuming that the absolute value of r is less than 1 for a geometric sequence, how can we figure out a formula for its corresponding infinite geometric series?0990

Well, we already figured out a formula that is true for any finite geometric series.0999

Remember: sn, the nth partial sum of any finite geometric series,1003

is a1 times (1 - rn)/(1 - r).1007

We just figured out that formula; that is pretty cool.1011

Not only that, but we also know that, as we look at partial sums containing more and more terms,1013

as a partial sum has more and more terms, they have to be growing closer and closer to the value that the infinite series will converge to.1018

As we put in more and more terms into our partial sum, it has to be getting closer to this value that it is going to converge to.1027

Think about why that is: if the partial sums were not getting closer to a specific value,1032

if they were moving around away from the specific value,1036

then it couldn't be converging to that, because it would always be changing around.1039

If it is going to converge to a single value, it has to be working its way towards it.1042

If it is working its way towards it, it must always be getting closer to the thing.1046

If it wasn't always getting closer, if it was sometimes jumping away, it wouldn't be working its way towards it; it would be going somewhere else.1049

Since we know that it is converging, we know that it must be working its way, as we have more and more terms in our partial sum.1054

As we put in more terms in our partial sum, we will be closer to the value that we are converging on.1065

As we add many more and more and more terms in our partial sum, we are going to be closer to the thing that we are converging to.1069

What we are asking ourselves is, "As we have really large values for n, what value are we getting close to?"1076

What happens to the formula we figured out, our nth partial sum formula,1082

sn = a1 times (1 - rn)/(1 - r), as the number of terms we have, our n, goes off to infinity?1086

As the value for the number of terms we have, our n, becomes infinitely large, what will happen to this formula?1093

Whatever happens to this formula is what we have to be converging to, because of the argument1098

that we just talked about, about how it has to be getting closer as we put in more and more terms.1102

Notice: the only term on the right side affected directly by the n is rn; there is no other term that directly has an n connected to it.1107

We can ask ourselves what happens to rn as our n grows very large.1115

Also, remember: r is less than 1; the absolute value of r has to be less than 1.1119

These two things combine as we ask n goes to infinity, and we are looking at our rn here,1125

since the absolute value of r is less than 1, we have that as n goes to infinity, rn has to go to 0.1133

So, it is going to shrink down to 0 as n grows infinitely large.1140

Why is that the case? Well, since the absolute value of r is less than 1, every step has to make it smaller.1144

For example, if we look at .9 raised to the 100, we get that that is less than .0001.1150

Why is this occurring? Well, since the absolute value of r is less than 1, we know that it is this fractional thing--1157

that effectively, every time we iterate it, every time we hit a term with this common ratio, it takes a little bite out of it.1164

Whether it is .1 or 1/2 or 3/5 or 922/1000, it is going to take a bite out of whatever the term that it is being multiplied against is.1170

As it takes infinitely many bites, since it is always shrinking it down, it means that it is always working its way towards this value of 0.1181

Infinitely many bites away gets us to having nothing.1189

Therefore, because rn is going to 0 as n goes to infinity, we have this part right here shrinking down to a 0 in our formula.1194

So, we get the following formula for an infinite geometric series: the infinite sum is equal to a1 times 1/(1 - r),1202

assuming that the absolute value of r is less than 1.1210

If the absolute value of r is greater than or equal to 1, we couldn't even talk about this in the first place,1212

because our series would be diverging, because it would always be growing and changing around on us.1217

But if we have that the absolute value of r is less than 1, all we need to know is our first term, a1, and the rate that it is growing at.1220

And we work it out through this formula, and we know what it will converge to over the long run.1228

All right, cool--we are ready for some examples.1232

The first one: Show that the sequence below is geometric; then give a formula for the general term (that is, the an, the nth term).1234

We have 7, 35, 175, what we want to ask ourselves is, "What number are we multiplying by each time?"1241

How do we get from 7 to 35? Well, we multiply by 5.1247

Let's check and make sure that that works: 35 to 175--yes, if we use a calculator (or do it in our heads,1250

or write it out by hand), we realize that, yes, we can multiply by 5 to get from there to there.1256

The same thing: 175 to 875: we multiply by 5; so we see that this is continuing.1259

Yes, it is geometric; that checks out.1263

Now, we want to give a formula for the general term.1266

We talked, in the lesson, about how an is equal to (let me write it the way we had it last time):1268

r, the rate that we are increasing at, to the n - 1, times a1.1276

What is our a1? Well, a1 is equal to 7, because it is the first term.1281

What is our r? r is equal to 5, since it is the number we are multiplying each time.1286

So, r = 5; r = 5; a1 = 7; so an is equal to 5n - 1 times 7.1290

And if we wanted to check this out, we could do a really quick check.1299

We could plug in...let's look at a2; that would be equal to 52 - 1 times 7,1303

so 5 to the 1 times 7; 5 times 7 is 35; we check that against what our second term was.1310

And indeed, that checks out; so it looks like we have our answer--there is our answer.1316

All right, the next example: Find the value of the finite geometric series below.1320

Notice: this does have an end--we stop at 3072, so it is not an infinite one.1326

If it were infinite, it would go out to infinity, so we wouldn't actually be able to find a value.1330

All right, how do we figure this out?1333

The first thing: what is the rate that we are increasing at?1336

To get from 3 to 6, we multiply by 2; to get from 6 to 12, we multiply by 2; so at this point, we realize that r equals 2.1338

What is the value a1? That is 3.1345

What we are looking for, remember: the formula we are going to do is: the nth partial sum, sn,1348

is equal to a1 times 1 minus the rate, raised to the nth power, divided by 1 minus the rate.1353

So, the only thing we have to figure out, that is left, is what our n is; n = ?.1359

How many value are we going to be at?1365

Here we are at a1, but this is a?; this is an, right over here.1367

What would that have to be? Well, we could set this up, using the formula that we talked about before, our general formula for the general term.1374

an is equal to rn - 1 times a1.1382

We plug in our an; that is 3072; 3072 =...our rate is 2, raised to the n - 1, times...a1 is 3.1388

So, we divide both sides by 3, because we are looking to figure out our n.1398

Divide both sides by 3, and we get 1024 = 2n - 1.1401

Now, you might just know immediately that 1024 is the tenth power of 2: 210 = 1024.1407

So, we would see that it is 10 steps to get forward; we would multiply 10 steps forward, so that would mean that our n is equal to 11.1415

We have figured out that to get 210...that is 10 steps; we multiplied all of them on the 3; we started at 31424

as our first stepping-stone; we stepped forward 10 the first stepping-stone, plus 10 steps forward,1430

means a total of 11 stepping-stones; so we have n = 11.1435

However, alternatively, we could just figure this equation right here out by using the work that we did with logarithms long ago in this class.1439

1024 = 2n - 1...well, what we can do is just take the log of both sides: log(1024) = log(2n - 1).1448

One of the properties of logs is that we can pull down exponents; that is why this is so useful.1459

We have n - 1 times log(2); log(1024) over log(2) equals n - 1.1463

log(1024)/log(2) is equal to 10; it equals n - 1, which tells us that n equals 11.1475

So, you could work this out just through raw algebra and using logarithms;1486

or you could work it out, if you recognized 210 as equal to 1024, if you just kept dividing 1024 with a calculator1489

until you saw how many steps it was; either way will end up getting us this value, that n equals 11.1495

All right, great; now we have everything that we need for our sum: sn, the nth partial sum1503

(in this case, the 11th partial sum of what this sequence would be) is going to be equal to...1509

a1 is 3 (our first term), times 1 minus our rate (it multiplied by 2 on each one of them),1517

raised to the 11th power (because the number of terms we have total is 11), divided by 1 minus the rate (2, once again).1525

We work this out: 3 times 1 - 211 is -2047; 1 - 2 is -1, so the -1 cancels out with the negative on top.1534

We have positive 2047 times 3; and we end up getting 6141; that is what we get once we add up all of those terms.1545

Great; the third example: Find the value of the below sum.1554

The sum is in sigma notation: from i = 0 up until 10 of 500 times 1/2 raised to the i minus 3 to the i divided by 64.1558

Previously, when we talked about sigma notation, series notation, we talked about how summations have properties1569

where we can separate some of the things in sigma notation, and we can pull out constants; great.1575

So, let's start by separating this: we can write this as Σ still has the same upper limit; the limits will not change...1580

the same index and lower limit of 500, times 1/2 to the what we are doing is separating around this subtraction...1587

minus...the series...same limits...of 3 to the i over 64.1599

So, we can separate, based on addition and subtraction, into two separate series.1607

Furthermore, we can also pull out constants and just multiply the whole thing.1610

We have 500 times the series; 10i equals 0, 1/2 to the i, minus...we pull out the 1/64, since that is just a constant, as well...1613

on the series, 10i equals 0; 3 to the

So, at this point, we can now use our formulas.1632

Remember: our formula was that the nth sum is equal to the first term, times 1 minus rn, over 1 - r.1634

So, each one of these is going to end up having different rates.1643

But they are going to end up having the same n.1646

Our n is going to be based off of going from 0, our starting index, all the way up to 10, our ending index.1648

What is our n if we go from 0 to 10?1656

Remember: we have to count that first step, so 0 up until 10 isn't 10; it is 11.1659

1 to 10 is 10, so 0 to 10 must be one more, 11; so we have n = 11.1664

All right, back into this: we have 500 times...let's use that formula...the series from i = 0 to 10 of 1/2 to the i...1670

well, what would it be for our a1--what would be the first term?1678

Well, if we plugged in i = 0 on (1/2)i, (1/2)0, or anything raised to the 0, is just 1; so our a1 is 1 there.1681

Times 1 minus...what is the rate? Well, we are multiplying by 1/2 each time, because it is 1/2 with an exponent on it.1690

So, 1/2 is our rate, raised to the n; our n that we figured out was 11, divided by 1 minus the same rate, 1/2.1695

Great; minus...over here, 1/64: we apply that formula again; so, the series from i = 0 to 10 of 3i:1705

what is our a1--what is the first term?1714

Well, 3 to the 0, because our starting index is 0 once again--the first term of this series would be 3 to the 0, because it is the first thing that would show up.1716

3 to the 0 is just 1 again; 1 minus...what is our rate? Our rate is 3, because we multiply by 3,1726

successively, for each iteration, because it is an exponent.1732

3 to the...what is our number of terms? 11; n = minus three.1735

Great; now we can work through a calculator to figure out what these things are.1741

We will save the incredibly boring actual doing of the arithmetic, but it shouldn't be too difficult.1744

This will become 500 times 2047, over 1024, minus 1 over 64, times 88573.1749

We can combine these things; actually, let's distribute our constants first.1765

2047...well, we will distribute just the multiplication; they are all constant numbers now.1769

255 thousand on our left, 875, divided by 256, minus 88573 over 64;1773

we have to put the right side on a common denominator, but once again, I will spare working out every single aspect.1793

This would simplify to -98417 divided by 256; and there is our answer, exactly precise.1799

If we wanted to, we could have approximated to decimals at some point, and we would get -384.44, approximately.1810

Notice: probably back here, when we were working at these steps, where we have these giant fractions,1819

there is a good chance, if you are working with a scientific calculator or a graphing calculator, that it doesn't show it perfectly in fractions.1824

You would end up getting fractional numbers; and for the most part, most teachers and textbooks would be fine with giving an answer in decimals instead.1829

So, either one of these two answers--the specific exact fraction, -98417/256, or approximately -384.44--both of them are perfectly fine.1835

The fourth example: Many lessons ago, when we first learned about exponential functions,1847

we had a story where a clever mathematician asked for one grain of rice on the first square of a chessboard,1851

then two on the second square, four on the third square, eight on the fourth square, 16, 32, 64, 128...1857

he keeps asking for doubling amounts of grains of rice.1872

So, if he had actually been paid on all of the squares, just as he was initially promised, how many grains of rice would he have total?1876

So, what we have here is a finite geometric series, because it is doubling each time.1885

What is our first term? Well, a1 was on the very first square; he had one grain, so our first term is going to be 1.1891

What is the rate that it increases at each time?1900

It goes 1, 2, 4, 8, it is doubling each time; so that means that the rate is multiplied by 2.1902

And what is the number of terms total? If we are going from the first square of a chessboard to...1908

a chessboard is 8 x 8, so the total squares on an 8 x 8 chessboard...we have 8 by 8, so that means we have 64 squares total.1913

So, that is going to be 64 times that this is going to happen; we have 64 terms here.1926

If we are going to figure this out, the nth partial sum, the 64th partial sum,1933

is going to be the first term, a1, times 1 minus the rate, 2, raised to the number of terms, 64, over 1 minus the rate, 2.1937

We work this out: we end up getting 1 - 264 over -1, which is the same thing as negative 1 - 264,1948

which we will end up seeing comes out to be very positive.1962

This is our precise answer for the number of grains; but let's see what that is approximately.1964

That comes out to be approximately 1.845x1019 grains of rice.1968

That is a big, incredibly huge number of grains of rice.1978

It might be a little bit hard to see just how many grains of rice that is: 1019--we are not used to working with scientific notation that large.1982

So, let's turn it into some words that we might understand.1989

That would be about the same as 18 quintillion (that is a quintillion: it goes million, billion, trillion, quadrillion, quintillion) grains of rice.1991

And we are, once again, not really used to working with numbers as incredibly large as the quintillion scale.2003

So, let's write that as 18 billion billion grains of rice; that is an incredible number of grains of rice--that is so many grains of rice!2008

How much rice is that? That is about more than the rice that has ever been produced by humanity, over all of humanity's time existing!2022

It might be approximately on the same scale as the amount of rice humanity has ever created.2030

I am talking about all of humanity ever creating this over the whole history of the world.2036

All of the rice that has ever been made...18 billion billion grains...that is probably somewhere on the same scale.2041

It might be a little bit more; it might be a little bit less; but that is the sort of scale.2046

All of the rice that humanity has ever created: we get to very, very large numbers very quickly2049

with these exponential functions when we are working in geometric stuff; that is something to keep in mind.2054

18 billion billion grains is as much rice as humanity has ever made--pretty amazing.2058

The final example: a super ball is dropped from a height of 3 meters.2065

On every bounce, it bounces 4/5 of the previous height.2068

If allowed to bounce forever, what is the total back-and-forth distance that the ball travels over?2071

Notice that it is total, so it is up and down.2076

Let's draw a picture to help ourselves see what is going on here.2079

We start...we have this ball; we drop the ball, and it falls three meters.2081

Then, it bounces; it is a super ball, so it bounces up, and it will go up by how much?2087

It will go up by 3 times 4/5, because it goes up to 4/5 of its previous height on every bounce.2092

Now, it is going to fall down; well, what height did it just fall down from?2099

It is going to fall down from the same thing, 3 times 4/5.2102

Then, it is going to bounce up again; it is going to bounce up by how much?2106

Well, it is going to bounce up to 3 times 4/5 times 4/5, because it is 4/5 of the previous height it came from.2108

3 times 4/5 times 4/5...well, we could just write that as 3 times (4/5)2.2113

Then, it is going to fall down, once again, from that same height; so it is 3 times (4/5)2.2118

And then, it is going to just continue on in this manner of going a little bit less each time:2124

3 times (4/5)3, and then down again: 3 times (4/5)3, and just continuing on in this manner forever.2129

We want to figure out the total back-and-forth distance that the ball travels over.2140

Notice: what we have here is ups and downs.2144

So, I am going to go with calculating the ups first: how much is from this up plus this up plus this up, going on forever and ever and ever and ever?2147

Notice: what that means we are dealing with is an infinite sum, because we are saying,2158

"If it were to continue bouncing forever, what would be the amount of distance it would travel over?"2162

That is equal to the first value, times 1 over 1 minus the rate; that was what we figured out that the infinite sum comes out to be.2167

It is the first value, times 1 over 1 minus the rate.2176

All right, in this case, for our up bounces, all of the ups are going to end up being...our first value2179

is a1, so our first value is this one right here, 3 times 4/5.2194

We might be tempted to think that it is 3 meters, but remember: this is all of the ups--we are looking at the ups first.2198

And we will see why at the end--why I chose to look at the ups first.2203

So, all ups is going to be...3 times 4/5 is our first value that we end up having occur.2207

And then, that is going to be times 1 over 1 minus...what is the rate? 4/5, because it is 4/5 on every bounce.2213

It goes up to 4/5 of its previous height, so the rate for the next one would be 4/5 of that, and 4/5 of that, and 4/5 of that.2221

Notice: there was one other requirement on being able to use this.2227

The absolute value of our rate must be less than 1.2230

But since it is 4/5, that ends up checking out; OK.2233

Keep going with this: that is 3 times 4/5; let's write it as 12/5; 12/5 times 1 over...1 - 4/5 is 1/5;2237

well, 1 divided by 1/5 is 12/5 times...1 over 1/5 is 5; so we have 5 times 1/5 on the bottom;2251

that cancels out; so we have 12 meters total of bouncing for our up values.2259

OK, but that is only part of it; now we also have to figure out what all of the down values are--2267

what is the down value here? the down value here? the down value here?...going out this way infinitely.2273

Well, we also have this part right here; but notice: all of these green things end up having a matching up value.2279

They each match up: the purple and the green, the purple ups and the green downs; they all match up.2292

The only one who is out of the normal case of matching up is that red first drop down.2297

So, what that means is that we can match all of our downs; they have a matching 12 meters,2303

because it matches all of the ups; but then we just have to add on the initial 3-meter drop at the beginning.2311

So, how many downs do we have in total?2319

Well, that 3-meter drop, plus the matched value (because it matches to the 12-meter value for all of the ups),2321

12 meters, plus 3: we get 15 meters, so the total number is going to be equal to the 12 from our ups,2327

plus the 15 from our downs; 12 + 15 means 27 meters in total.2345

Great; there we are--we finished that one.2354

All right, now we have a pretty good understanding of how sequences work.2355

We have worked through geometric sequences and arithmetic sequences.2358

We understand how series work; we have a pretty good idea of how sequences work; great.2361

All right, we will see you at later--goodbye!2365