Finding and Classifying Geometric Sequences

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  • 0:13 GS and Views of Cat Videos
  • 2:48 Creating the Formula
  • 5:28 Standard Formula of a GS
  • 6:19 Finding a GS Using Terms
  • 8:45 Summary
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Lesson Transcript
Instructor: Eric Garneau
Want your YouTube video to get a lot of hits? Besides including a cute baby or an adorable cat, getting your video to have a big common ratio is the key. Learn what I'm talking about here!

Geometric Sequences of Views of Cat Videos

2x3^5 represents the 6th term in this sequence
Geometric Sequence Pattern Condensing

A sequence is just a pattern, and a geometric sequence is a pattern that is generated through repeated multiplication. 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 by starting with 5 and multiplying by 2 four times. 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. 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.

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. At the end of the day, we'd be the only two people that had seen it. But let's say that within the next day, 3 of my friends and 3 of her friends had watched it, making 6 new hits. Then if each of those people were able to get 3 more people, that would be 18 more views, and then if each of those people got 3 of their friends we're already up to 54 more views in 4 days (a4=54)! 5 days in and I get 162 new hits (a5=162), 6 days brings in 486 (a6=486)… now we're getting somewhere!

It's important to note that 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 can find now is how many new views I'd get on any specific day. Simply using the pattern can help us figure out how many hits I'd be getting in the first few days, but trying to use the pattern one entry at a time to figure out how many new hits I'd get after a couple weeks would not only be long and tedious, I'd also probably end up making a mistake somewhere along the way.

Formula for the number of new hits for any day n after the initial video posting
Geometric Sequence Formula Example

Creating a Formula of a Geometric Sequence

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.

Generalizing the pattern for any term, this means that for any day n after I posted the video, the number of new hits is 2x3^n-1. The 2 in front represents the beginning value 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 going to 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 gone viral! Soon there will be a meme based on his picture.

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