Which sequence can be defined by the recursive formula \( f(1)=4, f(n+1)=f(n)-1.25 \) for \( n \geq \) \( \begin{array}{l}1,-0.25,-1.5,-2.75,-4, \ldots \\ 1,2.25,3.5,4.75,6, \ldots \\ 4,2.75,1.5,0.25,-1, \ldots \\ 4,5.25,6.5,7.75,8, \ldots\end{array} \)

The sequence defined by the recursive formula \( f(1) = 4, f(n+1) = f(n) - 1.25 \) starts at 4 and decreases by 1.25 for each subsequent term. So, the next terms would be \( 4 - 1.25 = 2.75 \), then \( 2.75 - 1.25 = 1.5 \), and so on. Following this pattern, the sequence is \( 4, 2.75, 1.5, 0.25, -1, \ldots \). Interestingly, the concept of recursive sequences can be traced back to ancient mathematics. The Greeks, particularly mathematician Fibonacci, explored sequences that build upon previous terms in intriguing ways. Recursive sequences aren't just numbers on a page; they have real-world applications too! For instance, understanding population growth, predicting financial trends, or modeling natural phenomena can all utilize sequences. Fun fact: the famous Fibonacci sequence is actually a special case of a recursive sequence!