Write a formula for the general or term for each geometric sequence.
$0.8,0.32,0.128,0.0512,\dots$

• A. $a_n=0.8{\left(2.5\right)}^{n-1}$
• B. $a_n={\left(0.4\right)}^{n-1}$
• C. $a_n={\left(2.5\right)}^{n-1}$
• D. $a_n=0.8{\left(0.4\right)}^{n-1}$

Answer: (D)

Verified step by step guidance

1. Identify the first term of the geometric sequence, which is the initial value given. In this sequence, the first term \(a_1\) is 0.8.
2. Determine the common ratio \(r\) by dividing the second term by the first term: calculate \(r = \frac{0.32}{0.8}\).
3. Recall the general formula for the \(n^{\operatorname{th}}\) term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\).
4. Substitute the values of the first term \(a_1 = 0.8\) and the common ratio \(r\) found in step 2 into the formula to write the explicit formula for \(a_n\).
5. Express the formula clearly as \(a_n = 0.8 \left(0.4\right)^{n-1}\), which represents the \(n^{\operatorname{th}}\) term of the given geometric sequence.