Write the recursive and explicit formulas for an arithmetic sequence with a1 = 4 and d = 5?

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An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a constant, called the common difference, to the previous term. In this example, the first term a1 is 4 and the common difference d is 5. The sequence begins with 4, and we add 5 to get each subsequent term. So the first five terms are 4, 9, 14, 19, and 24. Each term increases by 5 from the previous term. A recursive formula allows us to find any term in the sequence if we know the previous term. For our sequence with first term 4 and common difference 5, we start with a1 equals 4. Then a2 is a1 plus 5, which is 9. Similarly, a3 is a2 plus 5, which is 14. This pattern continues, so the recursive formula is: a1 equals 4, and an equals an-1 plus 5 for n greater than or equal to 2. This means to find any term, we take the previous term and add 5. An explicit formula lets us find any term directly without calculating all the previous terms. Let's analyze the pattern in our sequence. The first term a1 is 4, which we can write as 4 plus 0 times 5. The second term a2 is 9, which is 4 plus 1 times 5. The third term a3 is 14, which is 4 plus 2 times 5. The fourth term a4 is 19, which is 4 plus 3 times 5. We can see that the number in parentheses is always one less than the term number. So for the nth term, we have an equals 4 plus (n minus 1) times 5. Simplifying this, we get an equals 5n minus 1. Let's verify our formulas by calculating specific terms. To find the 5th term using the recursive formula, we need to calculate each term step by step. a1 is 4, a2 is 9, a3 is 14, a4 is 19, and a5 is 24. Using the explicit formula, a5 equals 5 times 5 minus 1, which is 25 minus 1, or 24. Both methods give the same result. For finding the 20th term, the recursive method requires calculating all previous terms, while the explicit formula gives us a20 equals 5 times 20 minus 1, which is 99. The explicit formula is clearly more efficient for finding distant terms, while the recursive formula shows the relationship between consecutive terms more clearly. Arithmetic sequences have many practical applications. Consider this problem: A savings plan starts with 4 dollars, and 5 dollars are added each week. How much money will there be after 12 weeks? Using our explicit formula an equals 5n minus 1, we find a12 equals 5 times 12 minus 1, which is 59 dollars. In which week will there be 49 dollars? We set 5n minus 1 equal to 49, solve for n, and find n equals 10. So there will be 49 dollars in week 10. The recursive formula is useful for understanding how the sequence grows step by step, while the explicit formula is more efficient for calculating distant terms directly.