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 first 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 2 plus a n minus 1. But we're actually given what the first two terms of the sequence are. So in order to write the first six terms, I'm actually just going to list them all out. We have a 1, which just equals 1 that's given to us, a 2, which equals also 1, and then we're gonna figure out a 3, a 4, a 5, and then a 6. And we're gonna use this, and we're gonna calculate these things by using the recursive formula that's given to us. So let's go ahead and get started here. The first two are just given to us. The third one is where things get a little bit interesting. Alright? So remember that the index is 3 here. Right? This is n equals 3. That's the 3rd 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 of the term that was 2 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 3 term, we're gonna have to take a, and we're gonna have to do 3 minus 2. We're gonna have to figure out that term, plus a and then 3 minus 1. Alright? So if you work this out, what you're gonna end up getting here is that this first term is really just a 1. So you're gonna have to add a 1 to a 2. So you're basically just adding the 2 terms that went prior to this term, and you're just gonna add them together. So in other words, a 1 plus a 2, and both of those things actually just end up being ones. So in other words, 1 plus 1 is 2. Alright? So those are our first three numbers, a one equals 1, a two equals 1, a three equals 2. And now we're gonna have to calculate a 4. Alright? So we're just gonna follow the same exact pattern. So this is the a 4. We're gonna have to take a, f 4 minus 2, some other words, the second term, and add it to a 4 minus 1. So in other words, we're just gonna do a2 +a3. Now we already know what a2 is. A2 is just 1. And then what's a3? Well, we actually just calculated that a3 is just 2. So this is just going to be 1+2, and that means that a4 is equal to 3. So there's lots of numbers going around because I'll be using subscripts and also numbers. But, basically, it's 112, and then 3. Let's calculate the 5th term in the sequence. Remember, this is n equals 5. So in order to calculate this, this is gonna be a we're gonna do 5 minus 2 plus a 5 minus 1. So this is gonna be a 3 plus a 4, and so on and so forth. Let's just go ahead and set up the 6th one. This is gonna be a 6 minus 2, plus a 6 minus 1. So this is now gonna be a 4 plus a 5. So you hopefully see that there's a pattern going on here. So it's a 12, a 23, a 34, a 45. And now we just plug in these numbers here. Alright. So 3 and a a 3 and a 4 is gonna be 2 plus 3. So, it's gonna be the 2 over here plus the 3 over here. So, this is gonna be 2 plus 3, and that's gonna equal 5. So, that is our 5th term is a 5 equals 5. Alright. So now we can figure out the last term over here, our 6th term, which is just gonna be a 4 plus a 5. Well, a 4 is just the 3. This is gonna be 3. And then, a 5 is gonna be the 5. And so, if you add these things together, you're actually gonna get 8. 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 112358, 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 }

20. Sequences, Series & Induction

Sequences

Precalculus

20. Sequences, Series & Induction

Sequences

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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 given: a_n = a_{n-2} + a_{n-1}. This means each term in the sequence is the sum of the two preceding terms.

Identify the initial conditions provided: a_1 = 1 and a_2 = 1. These are the first two terms of the sequence.

Calculate the third term using the recursive formula: a_3 = a_1 + a_2. Substitute the known values to find a_3.

Calculate the fourth term: a_4 = a_2 + a_3. Use the values of a_2 and the previously calculated a_3.

Continue this process to find the fifth term: a_5 = a_3 + a_4, and the sixth term: a_6 = a_4 + a_5, using the recursive formula and the values of the preceding terms.

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