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

Did you know that this recursive sequence defined by \( a_n \) produces a simple arithmetic progression? Starting at 3 for \( n=1 \), each subsequent term increases by 0.5. So, if you were to calculate a few terms, you'd find \( a_2 = 3.5 \), \( a_3 = 4 \), and so on. This kind of sequence is often used in real-world contexts, like in finance for interest calculations where the principle increases incrementally. When trying to work with recursive sequences like this one, a common mistake is neglecting to clearly define base cases. Always remember that without the first term, it can be challenging to derive further values! Ensure you track your terms carefully to avoid confusion. With a little practice, you’ll be able to handle these sequences like a pro!