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 \(r = \frac{0.32}{0.8}\) to find the factor by which the sequence is multiplied each time.
3. Recall the general formula for the \(n^{\operatorname{th}}\) term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio.
4. Substitute the values of \(a_1\) and \(r\) into the formula. This means replacing \(a_1\) with 0.8 and \(r\) with the value found in step 2, resulting in \(a_n = 0.8 \times (r)^{n-1}\).
5. Write the final formula explicitly with the numerical value of \(r\) to express the \(n^{\operatorname{th}}\) term of the sequence.