Question

Enter a recursive rule and an explicit rule for the arithmetic sequence. Then, find the 20th term of the sequence.

3, 0, -3, -6,...
The recursive rule is f(1) = ____.
f(n) = f(__) + ______.
The explicit rule is f(n) = ______.
f(20) = ______.

Answer 1

The recursive rule for the arithmetic sequence is f(1) = 3, f(n) = f(n-1) - 3, and the explicit rule is f(n) = 6 - 3n. The 20th term of the sequence is -54.

Step-by-step explanation:

To determine the recursive and explicit rules for the arithmetic sequence provided (3, 0, -3, -6,...), we first look at the pattern of the sequence. We can see that each term subtracts 3 from the previous term, which gives us a common difference of -3. Given this information, the recursive rule can be defined as f(1) = 3, and f(n) = f(n-1) - 3 for n > 1. For the explicit rule, we use the formula for an arithmetic sequence: f(n) = a + (n-1)d, where 'a' is the first term, and 'd' is the common difference. Here, f(n) = 3 + (n-1)(-3), which simplifies to f(n) = 3 - 3n + 3, and further simplifies to f(n) = 6 - 3n.

To find the 20th term, simply plug 20 into the explicit formula: f(20) = 6 - 3(20) = 6 - 60 = -54. Therefore, the 20th term of the sequence is -54.