\( a _ { n } = \left\{ \begin{array} { l l } { 3 } & { , n = 1 } \\ { a _ { n - 1 } + 0.5 } & { , n > 1 } \end{array} \right. \)

The given recurrence relation starts with \( a_1 = 3 \) and increases by 0.5 for each subsequent value of \( n \). So, if we calculate a few terms, we find \( a_2 = 3.5 \), \( a_3 = 4 \), \( a_4 = 4.5 \), and so on. This pattern reveals that \( a_n \) can be expressed explicitly as \( a_n = 3 + 0.5(n - 1) \), showcasing a linear relationship that makes it easy to predict future values! In real-world situations, this type of sequence can model scenarios where an initial value increases at a consistent rate over time. For instance, if you’re tracking the growth of a plant that grows half an inch each week, starting from 3 inches, you would use a similar mathematical approach to predict its height over the weeks!