Write a recursive formula for the geometric sequence {18,6,2,23,…}\left\lbrace18,6,2,\frac23,\ldots\right\rbrace{18,6,2,32,…}.

a n = a n − 1 3 a_{n}=\frac{a_{n-1}}{3} a n = 3 a n − 1

Write a recursive formula for the geometric sequence { 18 , 6 , 2 , 2 3 , … } \left\lbrace18,6,2,\frac23,\ldots\right\rbrace { 18 , 6 , 2 , 3 2 , … } .

Everyone. Welcome back. So in this practice problem, we're gonna write a recursive formula for a geometric sequence that's given to us down here below. So the sequence of numbers is eighteen, six, two, two thirds, and then so on and so forth. Let's get started here. Now remember that a geometric sequence is different from arithmetic because you're not looking at the difference between numbers. That number is constantly changing. 18 to six, it's 12. From six to two, it's negative four. From two to two thirds, it's gonna be some weird fraction. You're not looking at the difference between numbers, but the ratio. Alright? And once you find that ratio, we can find a recursive formula for this geometric sequence by this equation that we've seen before, which basically says that the next term in the sequence is equal to the previous term, a n minus one, times the common ratio r. So it's times r, not plus d this time. Alright? So let's take a look at the sequence here. I'm going from 18 to six. What do I have to multiply by? Well, unfortunately, what happens is that the numbers are getting a little bit smaller. But from 18 to six, you actually have to divide by three. Right? In other words, dividing by three is the same thing as multiplying by one third. Right? That's what you're multiplying by to get to the next number in the sequence. Ratios could be either whole numbers or they could be fractions. It's totally fine. So from six to two, what do I have to do? I also have to divide by three or, in other words, multiply by one third. And then if we get to two to two thirds, I also have to multiply by one third over here. Alright? So the common ratio in this sequence is that r is equal to one third each time. So then how do I use this to build a recursive formula? Just take a look at this formula over here. Basically, what this says here is that a n, the next term in the sequence, and you're just gonna take the previous term in the sequence, a n minus one, and you're just gonna multiply one third. Very straightforward. Now another way you may see this written here, so I'm gonna write or, is you may see this written as a n equals a n minus one divided by three. So you're just kind of combining the ratios together. Alright? So both of these are perfectly valid ways to express this recursive formula. Now remember, a recursive formula by itself is no good. You have to specify which term to start at. Otherwise, you don't know which first term to divide by three. So a one is equal to 18. Have to specify that. Otherwise, you can't recreate the numbers in the sequence. Alright? So that's your recursive formula. Thanks for watching, and I'll see you in the next video.

Write a recursive formula for the geometric sequence { 18 , 6 , 2 , 2 3 , … } \left\lbrace18,6,2,\frac23,\ldots\right\rbrace { 18 , 6 , 2 , 3 2 , … } .

a n = a n − 1 3 a_{n}=\frac{a_{n-1}}{3} a n = 3 a n − 1

a n = 3 a n − 1 a_{n}=3a_{n-1} a n = 3 a n − 1

a n = 18 a n − 1 a_{n}=18a_{n-1} a n = 18 a n − 1

a n = 2 3 a n − 1 a_{n}=\frac23a_{n-1} a n = 3 2 a n − 1

14. Sequences & Series

Sequences

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Sequences

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Multiple Choice

Write a recursive formula for the geometric sequence $\left\lbrace18,6,2,\frac23,\ldots\right\rbrace$ .

$a_{n}=3a_{n-1}$

$a_{n}=\frac{a_{n-1}}{3}$

$a_{n}=18a_{n-1}$

$a_{n}=\frac23a_{n-1}$

Verified step by step guidance

Step 1: Identify the type of sequence. The given sequence {18, 6, 2, 2/3, ...} is a geometric sequence because each term is obtained by multiplying the previous term by a constant ratio.

Step 2: Determine the common ratio (r). To find the common ratio, divide any term in the sequence by its preceding term. For example, r = 6 / 18 = 1/3.

Step 3: Write the recursive formula for the geometric sequence. A recursive formula expresses the nth term (a_n) in terms of the previous term (a_{n-1}). For a geometric sequence, the formula is generally written as a_n = r * a_{n-1}. Substitute the common ratio r = 1/3 into the formula.

Step 4: Include the initial term. To fully define the sequence, specify the first term (a_1). In this case, a_1 = 18.

Step 5: Combine the recursive formula and the initial term. The complete recursive formula for the sequence is: a_n = (1/3) * a_{n-1}, with a_1 = 18.

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