Question

Create 1 Arithmetic and 1 Geometric Sequence of your choice giving at least the first 4 terms of the sequence to show the pattern. Then for each sequence complete the following:

1. Identify the common difference (arithmetic) or common ratio (geometric).
2. Write the recursive formula for the sequence.
3. Write the explicit formula for the sequence.
4. Find f(10) for the sequence.

Answer 1

The arithmetic sequence starts at 2 with a common difference of 3 while the geometric sequence begins at 3 with a common ratio of 2. The respective formulas for finding the tenth term are f(n) = 2 + (n-1)*3 for arithmetic and f(n) = 3 * 2^(n-1) for geometric.

Step-by-step explanation:

For an arithmetic sequence, we could have the first four terms as 2, 5, 8, 11. The common difference is 3. The recursive formula could be written as f(n) = f(n-1) + 3 with f(1) = 2, and the explicit formula is f(n) = 2 + (n-1)*3. For f(10), you compute f(10) = 2 + (10-1)*3 = 29.

For a geometric sequence, let's take the first four terms as 3, 6, 12, 24. The common ratio is 2. The recursive formula is f(n) = f(n-1) * 2 with f(1) = 3, and the explicit formula is f(n) = 3 * 2^(n-1). To find f(10), calculate f(10) = 3 * 2^(10-1) = 3 * 512 = 1536.