Finding and Classifying Geometric Sequences

so a sequence is just a pattern, and a geometric sequence is a pattern that is generated through repeated multiplication.

that means that each new term is made by multiplying the previous one by the same thing over and over.

For example, we could have the finite geometric sequence 5, 10, 20, 40, 80 simply by starting with 5 and multiplying by 2.

Or we could get the infinite geometric sequence 8, 4, 2, 1, ½, ¼, … by beginning with 8 and repeatedly multiplying by ½, which ends up looking very similar to division.

As it turns out, we see these patterns all over the place.

now I’ve watched a lot of YouTube videos, and I’ve decided that my best bet to getting a really popular video is to get a cat and just follow it around with a camera 24/7.

It would be bound to do something cute eventually, especially if I put a goofy hat on it or something.

so within the first day of getting one of these funny videos I’d probably send it to my girlfriend, and then we’d both post it to our Facebook walls.

but at the end of the day, we’d be the only two people that had seen it.

But then maybe the next day, 3 of my friends and 3 of her friends had watched it, making 6 new hits on day 2.

and if that pattern able to continue each new person that sieze it, express it to the 3 of their friends, that would mean that the third day we’ve get 18 more views, then the forth day would be up to 54, the fifth day would be 162, by the 6 day will be getting 486 new views.

now we’re getting somewhere.

that was important to note that the pattern will making the sequence we’re making right now is only the number of new views I’m getting each day.

Figuring out how many total views we’ll reach is something we’re going to save for a later lesson.

What we are gonna focus on now is being able to determine how many new views I’d get on any specific day.

now simply using the pattern can help us figure out how many views I’d be get in the first couple of days, but trying to use the pattern one entry at a time to figure out how many hits I get say after a couple of weeks would not only be long and tedious, but i’m probably end up making a mistake somewhere along the way.

so what I’d rather have is a formula that will tell me how many new hits I get at any time simply by substituting in a number.

Looking back at the entries we came up with so far will help us to see the pattern that we can try to generalize into a formula.

We started at a1=2, then went up to a2=6 (2x3), then a3=18 (2x3x3), then 54 happened because we multiplied by 3 again (2x3x3x3).

I keep on multiplying by 3, so the next one is 2x3x3x3x3.

We can start condensing these multiplications by 3 into exponents, which means the sixth term would be 2x3^5.

Notice that the sixth term had an exponent of 5 on the 3.

If I condense the 3s into exponents on the fifth term, I get the 4th power; if I condense them on the fourth term, I get the 3rd power; the third term’s the 2nd; the second term’s the 1st; and the first term we can call the 0 power, because anything to the 0 power is 1, so we just get 2 x 1, or 2.

so taking this pattern in generalizing it for any term, gives us that the n term is 2x3^n-1 power.

The 2 represents were the pattern began a1, the 3 represents what would be called the ‘common ratio’ and the n-1 represents how many times we had to multiply by 3.

It’s n-1 because we didn’t get any new hits until the second day.

Now to figure out how many new hits I’m gonna get after two weeks is pretty straightforward.

I want to know how many new hits I’m going to get on the 14th day, so we substitute 14 in for n.

That gives us that a14=2x3^(14-1) power.

We do 3^13 power and we get 1,594,323.

We multiply that by 2 and it turns out that 14 short days after I posted the video I’m getting 3,188,646 new hits

My cat has officially went viral

there’s probably gonna be a meme about is soon or something like that.

what we have just come up with, is the general rule for the nth term of a geometric sequence

an=a1r^n-1 power.

This is actually pretty similar to an exponential function, only the letters have been changed around.

An is the general nth term, a1 is the first term, r is the common ratio, or the amount that we multiply by every step of the way.

It’s called the common ratio because if you divide any two consecutive terms you’ll get the same thing.

and lastly, the n in the n-1 power is whatever term you’re trying to find out.

so Apparently, if you can come up with a YouTube video that has a common ratio of 3, you’re on your way to internet stardom.

now once we get really good, we can come up with the rule to a geometric sequence simply by being given any two entries.

let’s say that we know that a3=9/4 and a6=243/32.

and we want to know what the rule is for the nth term.

I like drawing this out visually so we know what’s going on.

I don’t know what the first term or the second term is, but I do know that the third term is 9/4.

then the fourth term and the fifth term are unknown to me, but again the sixth term is 243/32.

now because it is a geometric sequence, I know that when I started with my third term (9/4) I simply multiplied by r three times in a row to get my sixth term, which means that 9/4r^3 = 243/32.

and now what I’ve set up is an equation that I can solve using inverse operations to figure out what the common ratio r must be.

I undo multiplication by a fraction by multiplying by its reciprocal, 4/9, that gives me 27/8 on the right hand side.

I can then undo a third power with a third root.

I take the cubed root of both sides and find that the cubed root of 27 is 3 and the cubed root of 8 is 2, and I find that our common ratio is 3/2.

The only other thing needed for our rule besides r is a1, which does mean that I have to work my way backwards to find a1.

again I can come back to my picture, divide by r.

you already go backward instead of multiplying.

I divide once by 3/2 and i get 3/2, and I divide one more time by 3/2 and find that a1 is simply equal to 1.

This means that my rule an=a1r^n-1 is an = 1 x 3/2^n-1 power.

now because multiplying by 1 doesn’t actually change anything, I can just condense my rule to an=3/2^n-1 power.

to reveiw a geometric sequence is a pattern of numbers generated by repeated multiplication, similar to exponential functions.

Because it’s similar to an exponential function, the general rule for a geometric sequence has an exponent in it, and we get an=a1r^n-1 power as the rule for the nth term of a geometric sequence.

This general rule can help us find terms of a geometric sequence that are far away from the beginning.