yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Using recursive formulas of geometric sequences | Mathematics I | High School Math | Khan Academy


2m read
·Nov 11, 2024

The geometric sequence ( a_i ) is defined by the formula where the first term ( a_1 ) is equal to -1/8 and then every term after that is defined as being so ( a_i ) is going to be two times the term before that. So, ( a_i ) is ( 2 \times a_{i-1} ).

What is ( a_4 ), the fourth term in the sequence? Pause the video and see if you can work this out.

Well, there's a couple of ways that you could tackle this. One is to just directly use these formulas. So, we could say that ( a_4 ), well, that's going to be this case right over here: ( a_4 ) is going to be equal to ( 2 \times a_3 ).

Well, ( a_3 ) if we go and use this formula is going to be equal to ( 2 \times a_2 ). Each term is equal to two times the term before it, and then we can go back to this formula again and say ( a_2 ) is going to be ( 2 \times a_1 ): ( 2 \times a_1 ).

And lucky for us, we know that ( a_1 ) is -1/8, so it's going to be ( 2 \times -1/8 ), which is equal to -1/4.

So, this is -1/4. Thus, ( 2 \times -1/4 ) is equal to -1/2 or -1/2.

So, ( a_4 ) is ( 2 \times a_3 ). ( a_3 ) is -1/2, so this is going to be ( 2 \times -1/2 ), which is going to be equal to -1.

So that's one way to solve it. Another way to think about it is: look, we have our initial term and we also know our common ratio. We know each successive term is two times the term before it.

So, we could explicitly write it as ( a_i ) is going to be equal to our initial term -1/8, and then we're going to multiply it by two. We're going to multiply it by ( 2^{i-1} ) times.

So, we could say, times ( 2^{i-1} ). Let's make sure that makes sense. So ( a_1 ) based on this formula ( a_1 ) would be -1/8 times ( 2^{1-1} ) which is ( 2^0 ). So that makes sense; that would be -1/8 based on this formula.

( a_2 ) would be -1/8 times ( 2^{2-1} ) so ( 2^1 ). So we're going to take our initial term and multiply it by 2 once, which is exactly right.

( a_2 ) is -1/4. So if we want to find the fourth term in the sequence, we could just say, well, using this explicit formula, we could say ( a_4 ) is equal to -1/8 times ( 2^{4-1} ), which is -1/8 times ( 2^3 ).

So this is equal to -1/8 times 8.

Thus, -1/8 times 8 equals -1.

And you might be a little bit tossed up on which method you want to use, but for sure this second method right over here where we come up with an explicit formula once we know the initial term and we know the common ratio—this would be way easier if you were trying to find, say, the 40th term.

Because doing the 40th term recursively like this would take a lot of time and frankly a lot of paper.

More Articles

View All
My All-Time FAVORITE Credit Cards
Lots of you guys, it’s great here! So, I realize I’ve been slipping up a lot lately. I’ve let a lot of you guys down, and this is just unacceptable. And that’s because it’s been way too long since I’ve made a credit card video—four months ago, to be exact…
Julia Hartz at Female Founders Conference 2014
Our next speaker is Julia Hearts. She’s the founder of Eventbrite, which is such a big deal. I know you’ve all used Eventbrite, and I’ve known Julia for years. Actually, we kind of started our companies around the same time, and I’ve wanted for years to p…
State checks on the judicial branch | US government and civics | Khan Academy
In previous videos, we had talked about the 1896 Supreme Court case Plessy versus Ferguson, which is a good one to know in general if you’re studying United States history and/or United States government. But this is where we got the principle of separate…
The 7 BEST Side Hustles To Start ASAP
What’s up, Graham? It’s guys here! So, you know what’s cooler than having one source of income? Seven different sources of income! And the average millionaire tends to agree with this too, at least according to their tax returns. Now seriously, the IRS a…
In Your Face - Mind Field (Ep 7)
If I asked you to show me a picture of your mother, you wouldn’t show me a, uh, closeup shot of her elbow. But you could, and you’d be right. That would be a photo of her, but it wouldn’t feel right because it’s not her face. That’s how important faces ar…
Watch: How Animals and People See the World Differently | National Geographic
[Music] What most people think of when they look at the world, they think other animals probably see the world pretty much the same way. Only with time do we realize that, of course, other animals don’t see the same things we see. That takes us to a sort …