Short Answer 
The common ratio in the geometric sequence from a1 to a4 is determined to be frac{5}{8}, calculated through the ratios a2 to a1, a3 to a2, and a4 to a3, with each step simplifying to the same ratio.
Step 1: Calculate the Ratio of a2 to a1
Begin by finding the ratio of a2 to a1. This is calculated as follows:
- Apply the formula: frac{a2}{a1} = frac{480}{768}.
- Simplify the fraction by dividing both the numerator and the denominator by 96: frac{480}{768} = frac{5}{8}.
- Use the ratio to express a2 in terms of a1: a2 = a1 * frac{5}{8}.
Step 2: Calculate the Ratio of a3 to a2
Next, find the ratio of a3 to a2 using a similar process:
- Apply the formula: frac{a3}{a2} = frac{300}{480}.
- Simplify this fraction by dividing both parts by 60: frac{300}{480} = frac{5}{8}.
- Express a3 in terms of a2: a3 = a2 * frac{5}{8}.
Step 3: Calculate the Ratio of a4 to a3
Finally, determine the ratio of a4 to a3 to complete the sequence:
- First, recognize a4 as 187.5 or frac{375}{2}.
- Calculate the ratio: frac{a4}{a3} = frac{frac{375}{2}}{300}, which simplifies to frac{375}{600}.
- Simplify this ratio by dividing by 75: frac{375}{600} = frac{5}{8}.
- Finally, express a4 in terms of a3: a4 = a3 * frac{5}{8}.
In conclusion, the common ratio in this geometric sequence is frac{5}{8}.