About this video
This video finds the first four terms of the recursive sequence f(n) = 1/5 times f(n - 1) with f(1) = 50, getting 50, 10, 2 and 2/5. Plotting them shows exponential decay, the typical shape of a geometric sequence, in contrast to the straight-line pattern of an arithmetic sequence.
You'll be able to
- use a recursive rule with a multiplier to find terms of a geometric sequence.
- plot the terms of a sequence and describe the shape of the graph.
- explain why geometric sequences look exponential while arithmetic sequences look linear.
Key words
- Geometric sequence
- A sequence where each term is the previous term multiplied by the same number.
- Recursive definition
- A rule that defines each term using the term before it, plus a starting value.
- Exponential decay
- A pattern where values keep shrinking by the same factor, getting smaller and smaller.
- Arithmetic sequence
- A sequence where each term is the previous term plus or minus a fixed amount.
Transcript
- [Teacher] We're told a sequence is defined by f of n is equal to 1/5 times f of n minus 1. So each term, whatever the value of the function is there or the sequence is for that term, it's 1/5 times the previous term for each whole number n, where n is greater than 1.
And then, they also tell us, which we need, what the first term is. f of 1 is equal to 50. Now, the reason I have this graph here is what I want you to do is pause this video and figure out what the value of this function is for n equals 1, 2, 3, and 4. And then, we're gonna be able to graph this together and think about what that graph looks like and why.
All right, now let's do this together. And what I like to do is set up a table here where on this left column, I have n, and then over here I have f of n on the right column. And so, let's first start with n equals 1. Well, they already tell us that f of 1 is equal to 50.
And if we want, we can plot that. When N is equal to one, and I should say that this is the n-axis instead of the x-axis. When n is equal to 1, f of 1 is 50, right over here. So, let's call this the y is equal to f of n axis. All right, let's do the next one. I'll do that in red.
When n is equal to 2, well f of 2 is going to be equal to, it says it right over here, 1/5 times f of 2 minus 1, or 1/5 times f of 1. So I could just write that as 1/5 times f of 1, which is equal to 1/5 times 50, which is equal to 10. So when n is equal to 2, this is equal to 10, right over here.
f of 2 is 10. And then, now, let's go to when n is equal to 3. f of 3, I think you see the pattern here, is equal to 1/5 times f of 2. Times f of 2. We know what f of 2 is, it's 10, so it's equal to 1/5 times 10, which is equal to two. So when n is equal to 3, y is equal to f of n is equal to 2, which is right about there.
And then last but not least, I'll do it in orange, when n is equal to 4, f of 4 is equal to 1/5 times f of 3, the previous term, which is equal to 1/5 times 2, which is equal to 2/5. So that's less than 1, so it's gonna be just right above 0 like that. And so, we have graphed those four points and you might see an interesting pattern here.
You might say, "Hey, you know what, this looks a lot like exponential decay." And that's not a coincidence because remember, every term here, it's 1/5 times the previous term, so we're decaying. We're multiplying each successive term by 1/5. It's getting smaller and smaller and smaller.
But that's what we're seeing here when we're dealing with a geometric sequence. Now, in this particular scenario, we defined this geometric sequence recursively. Each successive term, we've defined it in terms of the previous term, and then we got a starting condition.
There's other ways to define a geometric series so that it is not recursive, but it's good to get exposure to this. So generally speaking, if you have each successive term is going to be some multiple of the previous term, here it's a multiple less than 1, it could be a multiple greater than one, you're going to have points that look like they're on some type of exponential curve.
If, on the other hand, you had an arithmetic sequence where each successive term is plus or minus some fixed amount of the previous term, then it will look more linear.
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