Write a recursive sequence that represents the sequence defined by the following explicit formula: a, start subscript, n, end subscript, equals, minus, 48, left parenthesis, one half, right parenthesis, start superscript, n, plus, 1, end superscript a n ​

To convert the explicit formula $a_n = -48 \left(\frac{1}{2}\right)^{n+1}$ into a recursive formula, we need to express $a_n$ in terms of $a_{n-1}$. Here's the process:

Explicit Formula:

$a_n = -48 \left(\frac{1}{2}\right)^{n+1}$

Recursion:

1. Use the relationship between consecutive terms: $a_n = r \cdot a_{n-1}$ where $r$ is the ratio between consecutive terms.
2. Compute $r$: From the explicit formula: $\frac{a_n}{a_{n-1}} = \frac{-48 \left(\frac{1}{2}\right)^{n+1}}{-48 \left(\frac{1}{2}\right)^n} = \frac{\left(\frac{1}{2}\right)^{n+1}}{\left(\frac{1}{2}\right)^n} = \frac{1}{2}$ So, $r = \frac{1}{2}$.

Recursive Formula:

$a_n = \frac{1}{2} a_{n-1}, \quad n \geq 1$

Initial Condition:

From the explicit formula for $n = 0$: $a_0 = -48 \left(\frac{1}{2}\right)^{0+1} = -48 \cdot \frac{1}{2} = -24$

Final Recursive Sequence:

$a_0 = -24, \quad a_n = \frac{1}{2} a_{n-1}, \quad n \geq 1$

Let me know if you want further clarification or have additional questions! Here are some related questions to explore:

1. How do we derive the ratio $r$ for other sequences?
2. How does a geometric sequence differ from other types of sequences?
3. Can we solve the explicit formula directly for any term $n$?
4. What happens if the initial condition $a_0$ changes?
5. How does the recursive formula behave as $n \to \infty$?

Tip: Recursive sequences are easier to compute iteratively, but the explicit formula is more efficient for calculating distant terms directly!