Questions: Determine whether the given information represents an arithmetic or geometric sequence. Then write the recursive and the explicit equation for each. 5. 2,4,6,8, ... a. Arithmetic or geometric? b. Recursive: c. Explicit:

Solution Steps

To determine whether the sequence is arithmetic or geometric, we need to check the differences between consecutive terms. If the difference is constant, it's arithmetic; if the ratio is constant, it's geometric. For the sequence \(2, 4, 6, 8, \ldots\), calculate the difference between terms. If it's arithmetic, the recursive formula will use the common difference, and the explicit formula will use the first term and the common difference.

To determine whether the sequence \(2, 4, 6, 8, \ldots\) is arithmetic or geometric, we calculate the difference between consecutive terms. The difference between the first two terms is \(4 - 2 = 2\). Similarly, the difference between the second and third terms is \(6 - 4 = 2\), and between the third and fourth terms is \(8 - 6 = 2\). Since the difference is constant, the sequence is arithmetic.

For an arithmetic sequence, the recursive formula is given by: \[ a_n = a_{n-1} + d \] where \(d\) is the common difference. Here, \(d = 2\). Therefore, the recursive formula is: \[ a_n = a_{n-1} + 2 \]

The explicit formula for an arithmetic sequence is: \[ a_n = a_1 + (n-1) \cdot d \] where \(a_1\) is the first term of the sequence. Here, \(a_1 = 2\) and \(d = 2\). Thus, the explicit formula is: \[ a_n = 2 + (n-1) \cdot 2 \]

Final Answer