Final answer:
A recursive function with the base case f(1) = 10 and recursive case f(n) = f(n-1) + 12 generates an arithmetic sequence starting at 10 and increasing by 12 for each subsequent term.
Step-by-step explanation:
A recursive function is defined by two parts: the base case and the recursive case. In the scenario given, we want a recursive function with a base case of f(1) = 10. This means when the function is called with an argument of 1, it returns 10. For any other positive integer n, that is greater than 1, the recursive case applies: f(n) = f(n-1) + 12. In this recursive case, the function calls itself with an argument of n-1 and adds 12 to the result.
This function effectively creates an arithmetic sequence where each term is 12 greater than the previous term, starting from 10. To illustrate, f(2) would be f(1) + 12, which is 10 + 12, giving us 22. Then, f(3) would be f(2) + 12, which is 22 + 12, resulting in 34, and so on.
This recursive structure continues indefinitely, incrementing by 12 for each subsequent term and can be represented in a program or a mathematical proof to demonstrate the inductive step from one function call to the next.