Write a recursive formula and an explicit formula for the following arithmetic sequence.
 -2​, 6​, 14​, 22​, 30​, ...

For the given arithmetic sequence \(-2, 6, 14, 22, 30, \dots\), let's determine the recursive and explicit formulas.

### 1. Recursive Formula
In an arithmetic sequence, each term is obtained by adding a common difference \(d\) to the previous term. To find \(d\), we can subtract the first term from the second term:
\[
d = 6 - (-2) = 8
\]

So, the recursive formula is:
\[
a_n = a_{n-1} + 8, \quad a_1 = -2
\]
where \(a_n\) is the \(n\)-th term, and \(a_1\) is the first term of the sequence.

### 2. Explicit Formula
The explicit formula for an arithmetic sequence is given by:
\[
a_n = a_1 + (n - 1)d
\]

Substituting \(a_1 = -2\) and \(d = 8\):
\[
a_n = -2 + (n - 1) \cdot 8
\]
Simplify the expression:
\[
a_n = -2 + 8n - 8 = 8n - 10
\]

So, the explicit formula for the sequence is:
\[
a_n = 8n - 10
\]

### Final Formulas:
- **Recursive Formula**: \(a_n = a_{n-1} + 8\), with \(a_1 = -2\).
- **Explicit Formula**: \(a_n = 8n - 10\).

Would you like further details or explanations on these formulas?

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Here are some related questions for further practice:

1. How would you find the 10th term using the explicit formula?
2. What is the common difference, and why is it essential for an arithmetic sequence?
3. How do you convert a recursive formula to an explicit formula?
4. Can you identify the type of sequence given only the recursive formula?
5. How would the formulas change if the common difference was negative?

**Tip:** For arithmetic sequences, always identify the common difference first; it simplifies finding both recursive and explicit formulas.

Write a recursive formula and an explicit formula for the following arithmetic sequence: -2, 6, 14, 22, 30, ...

Discover how to write a recursive and an explicit formula for the arithmetic sequence -2, 6, 14, 22, 30. Learn step-by-step how to identify the common difference and construct both types of formulas. Ideal for students in Grades 9-10.

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