Write a recursive formula for the arithmetic sequence.{8,2,−4,−10,…}\left\lbrace8,2,-4,-10,\ldots\right\rbrace{8,2,−4,−10,…}

a n = a n − 1 − 6 a_{n}=a_{n-1}-6 a n = a n − 1 − 6 ; a 1 = 8 a_1=8 a 1 = 8

Write a recursive formula for the arithmetic sequence. { 8 , 2 , − 4 , − 10 , … } \left\lbrace8,2,-4,-10,\ldots\right\rbrace { 8 , 2 , − 4 , − 10 , … }

sequence are 8, 2, negative 4, and negative 10. Let's go ahead and get started here. Now remember that an arithmetic sequence is a sequence where the difference between the numbers is the same. And if that's true, then we can write a recursive formula here, which just tells us that the next term, a n, is equal to the previous term, a n minus 1, plus whatever that common difference is between the numbers. So, really, all we need is we just need to figure out the common difference between the numbers, and we can write our formula. Very straightforward. So let's take a look here. In this sequence here, from 8 to 2, you have to subtract 6. Then from 2 to negative 4, you also subtract 6. And then from negative 4 to negative 10, you also subtract 6. So this is clearly an arithmetic sequence because the difference between the numbers is always the same. So how do we use this to write a recursive formula? Well, this just tells us here that in order to get the next term in the sequence, we just take the previous term in the sequence, and we just subtract 6. That's really all there is to it here. This is the recursive formula for the sequence. Now this equation by itself doesn't tell you really much because you have to figure out where to start. You have to indicate what's what's starting point or what starting number you are. If I just say, you know, well, to get the next number, subtract the previous number by 6, and I don't tell you which first number to start from, then you don't really know what those numbers are gonna be. So this equation by itself isn't the whole thing. You have to indicate what the first number is, and that number here is 8. So this is how you write the recursive formula. You say, to get the next number, take the previous number and subtract by 6, and the first number that you start has to be 8. Alright? So this is the recursive formula. Pretty straightforward. Let me know if you have any questions. Thanks for watching.

Write a recursive formula for the arithmetic sequence. { 8 , 2 , − 4 , − 10 , … } \left\lbrace8,2,-4,-10,\ldots\right\rbrace { 8 , 2 , − 4 , − 10 , … }

a n = a n − 1 − 6 a_{n}=a_{n-1}-6 a n = a n − 1 − 6 ; a 1 = 8 a_1=8 a 1 = 8

a n = a n − 1 − 10 a_{n}=a_{n-1}-10 a n = a n − 1 − 10 ; a 1 = 6 a_1=6 a 1 = 6

a n = a n − 1 − 6 a_{n}=a_{n-1}-6 a n = a n − 1 − 6 ; a 1 = 6 a_1=6 a 1 = 6

a n = a n − 1 − 10 a_{n}=a_{n-1}-10 a n = a n − 1 − 10 ; a 1 = 8 a_1=8 a 1 = 8

9. Sequences, Series, & Induction

Arithmetic Sequences

College Algebra

9. Sequences, Series, & Induction

Arithmetic Sequences

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Multiple Choice

Write a recursive formula for the arithmetic sequence.
$\left\lbrace8,2,-4,-10,\ldots\right\rbrace$

$a_{n}=a_{n-1}-10$ ; $a_1=6$

$a_{n}=a_{n-1}-6$ ; $a_1=6$

$a_{n}=a_{n-1}-6$ ; $a_1=8$

$a_{n}=a_{n-1}-10$ ; $a_1=8$

Verified step by step guidance

Identify the first term of the sequence, which is given as $ a_1 = 8 $.

Determine the common difference $ d $ of the arithmetic sequence. This can be found by subtracting the first term from the second term: $ d = 2 - 8 = -6 $.

Recognize that the recursive formula for an arithmetic sequence is generally given by $ a_n = a_{n-1} + d $, where $ d $ is the common difference.

Substitute the common difference $ d = -6 $ into the recursive formula: $ a_n = a_{n-1} - 6 $.

Combine the first term and the recursive formula to express the sequence: $ a_1 = 8 $ and $ a_n = a_{n-1} - 6 $.

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