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 , … }

Everyone. Welcome back. So in this practice problem, we're gonna write a recursive formula for the sequence that's shown to us here. And the numbers in the sequence are eight, two, negative four, 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 one, 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 eight to two, you have to subtract six. Then from two to negative four, you also subtract six. And then from negative four to negative 10, you also subtract six. 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 six. 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 six, and I don't tell you which 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 number is, and that number here is eight. So this is how you write the recursive formula. You say, to get the next number, take the previous number and subtract by six, and the number that you start has to be eight. 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

14. Sequences & Series

Sequences

Business Calculus

14. Sequences & Series

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

Step 1: Understand the problem. We are tasked with writing a recursive formula for the given arithmetic sequence {8, 2, -4, -10, ...}. A recursive formula expresses each term in the sequence as a function of the previous term.

Step 2: Identify the first term (a₁) of the sequence. From the sequence {8, 2, -4, -10, ...}, the first term is a₁ = 8.

Step 3: Determine the common difference (d) of the arithmetic sequence. The common difference is calculated by subtracting any term from the term that follows it. For example, d = 2 - 8 = -6 or d = -4 - 2 = -6. Thus, the common difference is d = -6.

Step 4: Write the recursive formula. In an arithmetic sequence, the recursive formula is given by aₙ = aₙ₋₁ + d, where aₙ₋₁ is the previous term and d is the common difference. Substituting d = -6, the formula becomes aₙ = aₙ₋₁ - 6.

Step 5: Include the initial condition. To fully define the recursive formula, we must specify the first term. The initial condition is a₁ = 8. Therefore, the complete recursive formula is aₙ = aₙ₋₁ - 6 with a₁ = 8.

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