Write the recursive and explicit formulas for an arithmetic sequence with a1 = 2 and d = 3?

视频信息

答案文本 复制

视频字幕 复制

An arithmetic sequence is a sequence where each term is obtained by adding a constant difference to the previous term. In this example, we have a sequence with first term a1 equals 2 and common difference d equals 3. The sequence begins 2, 5, 8, 11, 14, and so on. Each term is found by adding 3 to the previous term. A recursive formula defines each term based on the previous term. For our arithmetic sequence with a1 equals 2 and d equals 3, the recursive formula is: a1 equals 2, and a sub n equals a sub n minus 1 plus 3 for n greater than or equal to 2. This means we start with 2, then each subsequent term is found by adding 3 to the previous term. So a2 equals a1 plus 3 which is 5, a3 equals a2 plus 3 which is 8, and so on. An explicit formula allows us to find any term directly without calculating all previous terms. Let's observe the pattern in our sequence: a1 equals 2 which is 2 plus 3 times 0, a2 equals 5 which is 2 plus 3 times 1, a3 equals 8 which is 2 plus 3 times 2, and a4 equals 11 which is 2 plus 3 times 3. We can see that each term follows the pattern a sub n equals 2 plus 3 times n minus 1. This is the general explicit formula a sub n equals a1 plus d times n minus 1. Substituting our values a1 equals 2 and d equals 3, we get a sub n equals 2 plus 3 times n minus 1. Simplifying this expression gives us a sub n equals 2 plus 3n minus 3, which further simplifies to a sub n equals 3n minus 1. Let's verify both formulas give the same results for specific terms. We'll test with n equals 1, 5, and 10. Using the recursive method: a1 equals 2, and each subsequent term is found by adding 3 to the previous term. So a2 equals 5, a3 equals 8, a4 equals 11, and a5 equals 14. Using the explicit formula a sub n equals 3n minus 1: For n equals 1, we get 3 times 1 minus 1 equals 2. For n equals 5, we get 3 times 5 minus 1 equals 14. For n equals 10, we get 3 times 10 minus 1 equals 29. As we can see in our verification table, both methods produce identical results, confirming that our formulas are correct. When to use each formula? The recursive formula is useful when you know a current term and want to find the next term, or when working with sequences defined recursively. For example, if you know the 8th term is 23, you can find the 9th term using the recursive formula: a9 equals a8 plus 3 which equals 23 plus 3 equals 26. The explicit formula is better when you need a specific term far out in the sequence or want to calculate multiple terms quickly. For example, to find the 50th term, we use the explicit formula: a50 equals 3 times 50 minus 1 which equals 150 minus 1 which equals 149. This would require 49 steps with the recursive formula! Our comparison chart summarizes when each approach is most effective.