Recursion

Beyond the Fibonacci sequence, many arithmetic and geometric sequences can also be defined recursively using simple rules and base cases.

• An arithmetic sequence with initial term \( a \) and difference \( d \) can be defined recursively as \( A(0) = a \), \( A(n) = A(n - 1) + d \).
• A geometric sequence with initial term \( a \) and ratio \( r \) can be defined as \( G(0) = a \), \( G(n) = G(n - 1) \cdot r \), which models exponential growth or decay.

Recursive definitions clarify how each term depends directly on the previous one, making the structure of the sequence explicit.

Although closed formulas like \( A(n) = a + nd \) or \( G(n) = ar^n \) are more efficient, recursive definitions offer insight into the logic of step-by-step construction.