To determine the sum of the infinite series \(1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} + \dots\), we first identify the pattern in the numerators. The numerators are \(1, 5, 12, 22, 35, \dots\). This sequence can be generalized as \(\frac{n(n+2)}{2}\). Therefore, the general term of the series, \(T_n\), is given by \(T_n = \frac{n(n+2)}{2 \cdot 6^{n-1}}\).
The sum of this series, which is a mixed series and not purely geometric, is found using series transformation techniques.
The sum is calculated as follows:
\[ \sum_{n=1}^{\infty} \frac{n(n+2)}{2 \cdot 6^{n-1}} = \frac{288}{125} \]
Thus, the sum of the infinite series is \(\frac{288}{125}\).