Sure! I'll provide you with a set of problems where you create the arithmetic equations in both recursive and explicit forms for linear sequences. These problems will involve finding both the recursive and explicit forms of the equations given some initial terms or information.

Problem 1:

Given: The first term $a_1 = 5$ and the common difference $d = 3$.

1. Find the recursive equation for the sequence.
2. Find the explicit equation for the sequence.

Answer:

• Recursive form: $a_1 = 5, \quad a_n = a_{n-1} + 3 \quad \text{for } n > 1$

• Explicit form: $a_n = 5 + (n - 1) \times 3 = 5 + 3n - 3 = 3n + 2$

Problem 2:

Given: The first term $a_1 = -2$ and the common difference $d = -4$.

1. Find the recursive equation for the sequence.
2. Find the explicit equation for the sequence.

Answer:

• Recursive form: $a_1 = -2, \quad a_n = a_{n-1} - 4 \quad \text{for } n > 1$

• Explicit form: $a_n = -2 + (n - 1) \times (-4) = -2 - 4n + 4 = -4n + 2$

Problem 3:

Given: The first term $a_1 = 10$ and the common difference $d = 0$.

1. Find the recursive equation for the sequence.
2. Find the explicit equation for the sequence.

Answer:

• Recursive form: $a_1 = 10, \quad a_n = a_{n-1} + 0 \quad \text{for } n > 1$

• Explicit form: $a_n = 10 + (n - 1) \times 0 = 10$ (This sequence is constant.)

Problem 4:

Given: The first term $a_1 = 7$ and the common difference $d = -2$.

1. Find the recursive equation for the sequence.
2. Find the explicit equation for the sequence.

Answer:

• Recursive form: $a_1 = 7, \quad a_n = a_{n-1} - 2 \quad \text{for } n > 1$

• Explicit form: $a_n = 7 + (n - 1) \times (-2) = 7 - 2n + 2 = 9 - 2n$

Problem 5:

Given: The first term $a_1 = -5$ and the common difference $d = 5$.

1. Find the recursive equation for the sequence.
2. Find the explicit equation for the sequence.

Answer:

• Recursive form: $a_1 = -5, \quad a_n = a_{n-1} + 5 \quad \text{for } n > 1$

• Explicit form: $a_n = -5 + (n - 1) \times 5 = -5 + 5n - 5 = 5n - 10$

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Summary of Terms:

• Recursive form: An equation that defines the $n$-th term based on the previous term(s). Typically, it has the form:
  $a_n = a_{n-1} + d$ where $d$ is the common difference.

• Explicit form: An equation that expresses the $n$-th term directly as a function of $n$. For an arithmetic sequence, it usually takes the form: $a_n = a_1 + (n - 1) \times d$

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Would you like further details on any specific part, or do you have any questions about the above problems? Here's some related practice for you to consider:

1. How does the explicit formula change when the common difference is positive versus negative?
2. What happens to the recursive and explicit formulas if the sequence is constant (i.e., the common difference is 0)?
3. Can you find an explicit formula from a given recursive equation? How would you do that?
4. How do you calculate the nth term in both forms for a large value of $n$?
5. Can you determine whether a given sequence is arithmetic by analyzing its differences?

Tip:

When writing the recursive form, always ensure to specify the initial term. The common difference determines the pattern, but the starting point is critical for complete definitions!