Solved on Dec 15, 2023

Write a recursive formula for the arithmetic sequence $-38, -26, -14, -2, \ldots$ . Find $a_1$ and $a_n$ for $n \geq 2$ .

STEP 1

Assumptions
1. The given sequence is arithmetic, which means the difference between consecutive terms is constant.
2. The first term of the sequence is $-38$ .
3. We need to find a recursive formula, which defines each term based on its preceding term(s).

STEP 2

Identify the first term of the sequence, which is given as $-38$ .
$a_1 = -38$

STEP 3

To find the common difference, subtract the first term from the second term.
$d = a_2 - a_1$

STEP 4

Plug in the given values for $a_2$ and $a_1$ to calculate the common difference.
$d = -26 - (-38)$

STEP 5

Simplify the expression to find the common difference.
$d = -26 + 38$ $d = 12$

STEP 6

Now that we have the common difference, we can write the recursive formula. The recursive formula for an arithmetic sequence is given by:
$a_n = a_{n-1} + d \text{ for } n \geq 2$

STEP 7

Plug in the value of the common difference $d$ into the recursive formula.
$a_n = a_{n-1} + 12 \text{ for } n \geq 2$

STEP 8

Combine the expressions for $a_1$ and the recursive formula for $a_n$ to complete the answer.
$\begin{array}{l} a_{1}=-38 \\ a_{n}=a_{n-1} + 12 \text{ for } n \geq 2 \end{array}$
This is the recursive formula for the given arithmetic sequence.

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Write a recursive formula for the arithmetic sequence $-38, -26, -14, -2, \ldots$ . Find $a_1$ and $a_n$ for $n \geq 2$ .

STEP 1

Assumptions
1. The given sequence is arithmetic, which means the difference between consecutive terms is constant.
2. The first term of the sequence is $-38$ .
3. We need to find a recursive formula, which defines each term based on its preceding term(s).

STEP 2

Identify the first term of the sequence, which is given as $-38$ .
$a_1 = -38$

STEP 3

To find the common difference, subtract the first term from the second term.
$d = a_2 - a_1$

STEP 4

Plug in the given values for $a_2$ and $a_1$ to calculate the common difference.
$d = -26 - (-38)$

STEP 5

Simplify the expression to find the common difference.
$d = -26 + 38$ $d = 12$

STEP 6

Now that we have the common difference, we can write the recursive formula. The recursive formula for an arithmetic sequence is given by:
$a_n = a_{n-1} + d \text{ for } n \geq 2$

STEP 7

Plug in the value of the common difference $d$ into the recursive formula.
$a_n = a_{n-1} + 12 \text{ for } n \geq 2$

STEP 8

Combine the expressions for $a_1$ and the recursive formula for $a_n$ to complete the answer.
$\begin{array}{l} a_{1}=-38 \\ a_{n}=a_{n-1} + 12 \text{ for } n \geq 2 \end{array}$
This is the recursive formula for the given arithmetic sequence.

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Write a recursive formula for the arithmetic sequence −38,−26,−14,−2,…-38, -26, -14, -2, \ldots−38,−26,−14,−2,…. Find a1a_1a1​ and ana_nan​ for n≥2n \geq 2n≥2.

STEP 1

Assumptions1. The given sequence is arithmetic, which means the difference between consecutive terms is constant.2. The first term of the sequence is −38-38−38.3. We need to find a recursive formula, which defines each term based on its preceding term(s).

STEP 2

Identify the first term of the sequence, which is given as −38-38−38.a1=−38a_1 = -38a1​=−38

STEP 3

To find the common difference, subtract the first term from the second term.d=a2−a1d = a_2 - a_1d=a2​−a1​

STEP 4

Plug in the given values for a2a_2a2​ and a1a_1a1​ to calculate the common difference.d=−26−(−38)d = -26 - (-38)d=−26−(−38)

STEP 5

Simplify the expression to find the common difference.d=−26+38d = -26 + 38d=−26+38 d=12d = 12d=12

STEP 6

Now that we have the common difference, we can write the recursive formula. The recursive formula for an arithmetic sequence is given by:an=an−1+d for n≥2a_n = a_{n-1} + d \text{ for } n \geq 2an​=an−1​+d for n≥2

STEP 7

Plug in the value of the common difference ddd into the recursive formula.an=an−1+12 for n≥2a_n = a_{n-1} + 12 \text{ for } n \geq 2an​=an−1​+12 for n≥2

STEP 8

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Write a recursive formula for the arithmetic sequence $-38, -26, -14, -2, \ldots$ . Find $a_1$ and $a_n$ for $n \geq 2$ .

Assumptions
1. The given sequence is arithmetic, which means the difference between consecutive terms is constant.
2. The first term of the sequence is $-38$ .
3. We need to find a recursive formula, which defines each term based on its preceding term(s).

Identify the first term of the sequence, which is given as $-38$ .
$a_1 = -38$

To find the common difference, subtract the first term from the second term.
$d = a_2 - a_1$

Plug in the given values for $a_2$ and $a_1$ to calculate the common difference.
$d = -26 - (-38)$

Simplify the expression to find the common difference.
$d = -26 + 38$ $d = 12$

Now that we have the common difference, we can write the recursive formula. The recursive formula for an arithmetic sequence is given by:
$a_n = a_{n-1} + d \text{ for } n \geq 2$

Plug in the value of the common difference $d$ into the recursive formula.
$a_n = a_{n-1} + 12 \text{ for } n \geq 2$