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 18, 6, 2, 2 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 6, it's 12. From 6 to 2, it's negative 4. From 2 to 2 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 1, 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 6. 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 6, you actually have to divide by 3. Right? In other words, dividing by 3 is the same thing as multiplying by 1 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 6 to 2, what do I have to do? I also have to divide by 3 or, in other words, multiply by 1 third. And then if we get to 2 to 2 thirds, I also have to multiply by 1 third over here. Alright? So the common ratio in this sequence is that r is equal to 1 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 1, and you're just gonna multiply 1 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 an equals an minus 1 divided by 3. 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 3. So a 1 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

9. Sequences, Series, & Induction

Geometric Sequences

College Algebra

9. Sequences, Series, & Induction

Geometric 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

Identify the first term of the sequence, which is 18.

Determine the common ratio by dividing the second term by the first term: $ \frac{6}{18} = \frac{1}{3} $.

Verify the common ratio by checking subsequent terms: $ \frac{2}{6} = \frac{1}{3} $ and $ \frac{\frac{2}{3}}{2} = \frac{1}{3} $.

Write the recursive formula using the first term and the common ratio: $ a_1 = 18 $ and $ a_n = \frac{a_{n-1}}{3} $ for $ n \geq 2 $.

The recursive formula for the sequence is $ a_n = \frac{a_{n-1}}{3} $ with the initial condition $ a_1 = 18 $.

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