**Understanding Geometric Sequences****Problem Statement:**Let \(a_n\) denote the \(n^\text{th}\) term of a geometric sequence. Let \(a_0 = 512\) and \(a_{n+1} = 0.5a_n\). What is the explicit formula for this sequence?**Options:**1. \(a_n = 256(0.5)^n\)2. \(a_n = 512(0.5)^n\)3. \(a_n = 0.5(512)^n\)4. \(a_n = 512(n)^{0.5}\)### Detailed Explanation:For a geometric sequence, each term is derived by multiplying the previous term by a constant ratio. Here, the first term (\(a_0\)) is given as 512, and the ratio (\(r\)) is 0.5.The explicit formula for the \(n^\text{th}\) term of a geometric sequence is given by:\[ a_n = a_0 \cdot r^n \]Given:\[ a_0 = 512 \]\[ r = 0.5 \]Substitute these values into the formula:\[ a_n = 512 \cdot (0.5)^n \]Thus, the explicit formula for this sequence is:### **\(a_n = 512(0.5)^n\)****Solution:**Therefore, the correct option is:\[ \text{Option 2: } a_n = 512(0.5)^n \]This option matches the derived formula.

Name: **Understanding Geometric Sequences****Problem Statement:**Let \(a_n\
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: **Understanding Geometric Sequences****Problem Statement:**Let \(a_n\) denote the \(n^\text{th}\) term of a geometric sequence. Let \(a_0 = 512\) and \(a_{n+1} = 0.5a_n\). What is the explicit formula for this sequence?**Options:**1. \(a_n = 256(0.5