Write the first 6 terms of the sequence given by the recursive formula an=an−2+an−1a_{n}=a_{n-2}+a_{n-1}an=an−2+an−1 ; a1=1a_1=1a1=1 ; a2=1a_2=1a2=1.

{ 1 , 1 , 2 , 3 , 5 , 8 } \left\lbrace1,1,2,3,5,8\right\rbrace { 1 , 1 , 2 , 3 , 5 , 8 }

Write the first 6 terms of the sequence given by the recursive formula a n = a n − 2 + a n − 1 a_{n}=a_{n-2}+a_{n-1} a n = a n − 2 + a n − 1 ; a 1 = 1 a_1=1 a 1 = 1 ; a 2 = 1 a_2=1 a 2 = 1 .

Everyone. Welcome back. So in this practice problem, we're gonna write the six terms of a sequence. And it's given to us not by a general formula, but it's given to us with a recursive formula. Remember, a recursive formula is basically a formula that allows us to figure out the next term based on the terms that go before it. So in this, this recursive formula says that the next term a n is going to equal a n minus two plus a n minus one. And we're actually given what the two terms of the sequence are. So in order to write the six terms, I'm actually just going to list them all out. So we have a1, which just equals one, that's given to us, a2, which equals also one, and then we're going to figure out a3, a four, a5, and then a6. And we're going to use this, and we're going to calculate these things by using the recursive formula that's given to us. So let's go ahead and get started here. The two are just given to us. The one is where things get a little bit interesting. Alright? So remember that the index is three here. Right? This is n equals three. That's the term. So what does this formula say? This formula says that in order to calculate any nth term, we're actually gonna have to take the number for the term that was two terms prior, and we're gonna have to add it to the the term that came directly before that term. So that sounds really confusing here, but, basically, if you just go ahead and plug in n's for this, you'll see that it's actually pretty straightforward. To calculate the n equals three term, we're gonna have to take a, and we're gonna have to do three minus two. We're gonna have to figure out that term, plus a and then three minus one. Alright? So if you work this out, what you're gonna end up getting here is that this term is really just a one. So we're gonna have to add a one to a two. So you're basically just adding the two terms that went prior to this term, and you're just gonna add them together. So in other words, a one plus a two, and both of those things actually just end up being ones. So in other words, one plus one is two. Alright? So those are our three numbers, a one equals one, a two equals one, a three equals two. Now we're gonna have to calculate a four. Alright? So we're just gonna follow the same exact pattern. So this says that a four, we're gonna have to take a, a four minus two, some other words, the term, and add it to a four minus one. So in other words, we're just gonna do a two plus a three. Now we already know what a two is. A two is just one. And then what's a three? Well, we actually just calculated that a three is just two. So this is just gonna be one plus two, and that means that a four is equal to three. Alright? So there's lots of numbers going around because I'm using subscripts and also numbers. But basically, it's one, one, two, and then three. Let's calculate the term in the sequence. Remember, this is n equals five. So in order to calculate this, this is going to be a, we're going to do five minuteus two, plus a five minuteus one. So this is going to be a three plus a four, and so on and so Let's just go ahead and set up the one. This is going to be a six minuteus two, plus a six minuteus one. So this is now going to be a four plus a five. So you hopefully see that there's a pattern going on here. So it's a 12, a 23, a 34, A45. And now we just plug in these numbers here. Alright. So, three and a a3 and a four is going to be two plus three. So, it's going to be the two over here plus the three over here. So, this is going to be two plus three, and that's going to equal five. So, that is our term, is a five equals five. Alright. So, now, we can figure out the last term over here, our term, which is just going to be a four plus a five. Well, a four is just the three. This is going to be three. And then, a five is going to be the five. And so if you add these things together, you're actually gonna get eight. Now, this specific sequence here given by this recursive formula actually has a name. It's called the Fibonacci sequence. But you'll see that it goes one one, two three, five eight, where basically each one of these numbers here is made up of the two numbers that went prior added together. Alright? So that's it for this one, folks. Hopefully, you guys understand this, this, recursive formula, and I'll see you in the next video. Thanks for watching.

Write the first 6 terms of the sequence given by the recursive formula a n = a n − 2 + a n − 1 a_{n}=a_{n-2}+a_{n-1} a n = a n − 2 + a n − 1 ; a 1 = 1 a_1=1 a 1 = 1 ; a 2 = 1 a_2=1 a 2 = 1 .

{ 1 , 1 , 2 , 3 , 5 , 8 } \left\lbrace1,1,2,3,5,8\right\rbrace { 1 , 1 , 2 , 3 , 5 , 8 }

{ 1 , 1 , 2 , 3 , 5 , 18 } \left\lbrace1,1,2,3,5,18\right\rbrace { 1 , 1 , 2 , 3 , 5 , 18 }

{ 1 , 2 , 3 , 5 , 8 , 13 } \left\lbrace1,2,3,5,8,13\right\rbrace { 1 , 2 , 3 , 5 , 8 , 13 }

{ 0 , 1 , 1 , 2 , 3 , 5 } \left\lbrace0,1,1,2,3,5\right\rbrace { 0 , 1 , 1 , 2 , 3 , 5 }

14. Sequences & Series

Sequences

Calculus

14. Sequences & Series

Sequences

Struggling with Calculus?

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Multiple Choice

Write the first 6 terms of the sequence given by the recursive formula $a_{n}=a_{n-2}+a_{n-1}$ ; $a_1=1$ ; $a_2=1$ .

$\left\lbrace1,1,2,3,5,18\right\rbrace$

$\left\lbrace1,2,3,5,8,13\right\rbrace$

$\left\lbrace0,1,1,2,3,5\right\rbrace$

$\left\lbrace1,1,2,3,5,8\right\rbrace$

Verified step by step guidance

Start by understanding the recursive formula: a_n = a_{n-2} + a_{n-1}. This means that each term in the sequence is the sum of the two preceding terms.

The initial conditions are given as a_1 = 1 and a_2 = 1. These are the first two terms of the sequence.

To find the third term (a_3), use the formula: a_3 = a_1 + a_2. Substitute the values of a_1 and a_2 to calculate a_3.

To find the fourth term (a_4), use the formula: a_4 = a_2 + a_3. Substitute the values of a_2 and a_3 to calculate a_4.

Continue this process to find the fifth term (a_5 = a_3 + a_4) and the sixth term (a_6 = a_4 + a_5). This will give you the first six terms of the sequence.

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