Final Answer:
The recursive function that describes the arithmetic sequence with the first four terms 7, 13, 19, 25 is f(1) = 7, f(n) = f(n - 1) + 6, n ≥ 2.
Step-by-step explanation:
An arithmetic sequence is characterized by a common difference (d) between consecutive terms. In this case, the first four terms of the sequence are 7, 13, 19, and 25. To determine the recursive function, observe the pattern of the sequence. The common difference between each term is +6. Starting from the initial term f(1) = 7, each subsequent term can be obtained by adding 6 to the previous term. Thus, the recursive function is expressed as f(n) = f(n - 1) + 6 for n ≥ 2, where f(1) = 7.
Verifying this sequence using the recursive formula, when n = 2, f(2) = f(2 - 1) + 6 = f(1) + 6 = 7 + 6 = 13. Similarly, for n = 3, f(3) = f(3 - 1) + 6 = f(2) + 6 = 13 + 6 = 19, and for n = 4, f(4) = f(4 - 1) + 6 = f(3) + 6 = 19 + 6 = 25. Therefore, the given recursive function accurately generates the sequence's terms by adding 6 to the previous term starting from the initial value of 7.
This recursive function provides a formulaic representation of the arithmetic sequence, allowing for the calculation of any term in the sequence by recursively applying the rule of adding 6 to the preceding term. It serves as a helpful tool in predicting subsequent terms of the sequence without explicitly listing all the intermediate terms, showcasing the pattern observed in the sequence.