Hey, everyone. Welcome back. So we just finished talking a lot about arithmetic sequences, like, for example, 36912, where the difference between each number is always the same number. But let's take a look at this sequence over here, 392781. Clearly, we can see that the difference between numbers is never the same. It's constantly getting bigger. But there's still actually a pattern going on with this sequence. What I'm going to show you in this video is that this is a special type of sequence called a geometric sequence, and what we're going to see is that there's a lot of similarities between how we use the information and the pattern across the numbers to set up a recursive formula for these types of sequences. So I want to show you how to do that and the basic difference between these two types, and we'll do some examples. Let's get started. So remember that arithmetic sequences are special types where the difference between terms was always the same. For example, the common difference in this situation that the sequence was 3. A geometric sequence is a special type where the ratio between terms is always the same number. So, for example, from 3 to 9, you have to multiply by 3. From 9 to 27, you also multiply by 3. From 27 to 81, you multiply by 3. So instead of adding 3 to each number to get the next one, you have to multiply by 3 to get the next number. Now this ratio over here is called the common ratio, and the letter we use for this is little r. So little r in this case is equal to 3. Kind of like how in this case little d was equal to 3. Alright? Now we can use this common ratio to find additional terms by setting up a recursive formula. Remember, recursive formulas are just formulas that tell you the next term based on the previous term. So in this situation, we just took the previous term and added 3. Well, in this geometric sequence, we're going to take the previous term, and instead we have to multiply by 3. That's really all there is to it. The way that you use these formulas to find the next terms is exactly the same. Alright? So, in fact, this sort of general sort of structure that you'll see for these recursive formulas for geometric sequences is they'll always look like this.
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Geometric Sequences: Study with Video Lessons, Practice Problems & Examples
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Geometric sequences differ from arithmetic sequences in that they use a common ratio, denoted as r, to find subsequent terms by multiplying the previous term. The recursive formula is expressed as An = An-1r. The general formula for the nth term is An = A1r^(n-1). Understanding these formulas allows for efficient calculation of terms in geometric sequences.
Geometric Sequences - Recursive Formula
Video transcript
Write a recursive formula for the geometric sequence {18,6,2,32,…}.
Geometric Sequences - General Formula
Video transcript
Everyone, welcome back. So we just spent a lot of time talking about recursive formulas for arithmetic sequences, and then we saw how to write the general formula. Well, we just learned how to write a recursive formula for a geometric sequence that allows you to calculate terms based on the previous term. But now we're going to take a look at how to write the general formula, which allows you to calculate any term without having to know the previous one. And what we're going to see is there are a lot of parallels with how we did this for arithmetic sequences. I'm going to show you how to write a general formula for a geometric sequence, and then we'll do an example together. Let's get started here.
So, a general formula is going to give you the nth term. And just like the arithmetic sequence, it's going to be based on the first term. But now instead of the common difference
So with an arithmetic sequence, we saw that for a recursive formula, you're just doing
So here, we can again see lots of similarities between these two types of equations. So for example, in this sequence, 3, 6, 12, 24, all you need to write the general formula is the first term, which is this 3 over here, and then you just need the common difference. And again, this common difference, what we see here is that
This is the general formula for this sequence over here, whereas this was the recursive formula. Alright? Now, let's use this to actually calculate the 4th term, which we know should be 24. Well,
So this is the general sort of formats for a general formula. It's the first term times
So if we want to write a general formula, we're just going to start out with our general formula equation, which is that
Alright? So we have what our two terms are,
Find the msup msup msup 10th term of the geometric sequence in which a1=5 and r=2.
5,120
1,280
10,240
2,560
Write a formula for the general or msup msup msup nth term of the geometric sequence where a7=1458 and r=−3.
Here’s what students ask on this topic:
What is a geometric sequence and how does it differ from an arithmetic sequence?
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted as
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How do you find the common ratio in a geometric sequence?
To find the common ratio
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What is the recursive formula for a geometric sequence?
The recursive formula for a geometric sequence is used to find the next term based on the previous term. It is expressed as
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How do you write the general formula for the nth term of a geometric sequence?
The general formula for the nth term of a geometric sequence allows you to find any term without knowing the previous term. It is expressed as
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How do you find the 12th term in a geometric sequence?
To find the 12th term in a geometric sequence, you use the general formula
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