Find the sum of the following geometric series. \( 3+2.55+2.1675+1.842375+\ldots \)

To find the sum of the geometric series \( 3 + 2.55 + 2.1675 + 1.842375 + \ldots \), we first need to identify the first term and the common ratio. The first term \( a \) is \( 3 \). Next, we calculate the common ratio \( r \): \[ r = \frac{2.55}{3} \approx 0.85 \] We can verify this by calculating the ratio for other terms as well: \[ \frac{2.1675}{2.55} \approx 0.85 \quad \text{and} \quad \frac{1.842375}{2.1675} \approx 0.85 \] Now that we have identified the first term \( a = 3 \) and the common ratio \( r \approx 0.85 \), we can use the formula for the sum \( S \) of an infinite geometric series, which is given by: \[ S = \frac{a}{1 - r} \] Since \( |r| < 1 \), we can apply the formula: \[ S = \frac{3}{1 - 0.85} = \frac{3}{0.15} = 20. \] Thus, the sum of the geometric series is \( \boxed{20} \).