An arithmetic sequence is defined by a common difference ($d$) between consecutive terms. The recursive formula for an arithmetic sequence is generally:

$a_n = a_{n-1} + d$

Step 1: Determine the common difference ($d$)

The sequence is: $-4, -\frac{15}{2}, -11, -\frac{29}{2}, \ldots$

The difference between consecutive terms is: $d = a_2 - a_1 = -\frac{15}{2} - (-4) = -\frac{15}{2} + \frac{8}{2} = -\frac{7}{2}.$

Step 2: Write the recursive formula

Using $d = -\frac{7}{2}$ and the first term $a_1 = -4$, the recursive formula is:

$a_n = a_{n-1} - \frac{7}{2}, \quad \text{for } n \geq 2.$

Step 3: Find the 10th term

To find the 10th term ($a_{10}$), we use the explicit formula of an arithmetic sequence:

$a_n = a_1 + (n-1) \cdot d$

Substitute $a_1 = -4$, $d = -\frac{7}{2}$, and $n = 10$:

$a_{10} = -4 + (10-1) \cdot \left(-\frac{7}{2}\right)$

Simplify:

$a_{10} = -4 + 9 \cdot \left(-\frac{7}{2}\right)$

$a_{10} = -4 - \frac{63}{2}$

Convert $-4$ to a fraction with denominator 2:

$a_{10} = -\frac{8}{2} - \frac{63}{2}$

$a_{10} = -\frac{71}{2}.$

Final Answer:

The recursive formula is:

$a_n = a_{n-1} - \frac{7}{2}, \quad \text{for } n \geq 2, \quad \text{with } a_1 = -4.$

The 10th term is:

$a_{10} = -\frac{71}{2}.$

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Do you have any questions or need a breakdown of the steps? Here are five related questions to expand your understanding:

1. How do we derive the explicit formula for an arithmetic sequence from its recursive definition?
2. What is the difference between arithmetic and geometric sequences?
3. How would the formula change if the common difference was positive?
4. Can you graph this arithmetic sequence? What type of graph would it create?
5. How do we find the sum of the first 10 terms of this arithmetic sequence?

Tip: Always confirm the common difference by checking multiple consecutive terms in the sequence.