Final answer:
The series a_n = 4 - n/2 is an arithmetic series because the difference between consecutive terms is constant. The common difference of the series is -0.5.
Step-by-step explanation:
To determine if the series an = 4 - n/2 is arithmetic or geometric, we need to investigate the pattern of the terms. In an arithmetic series, the difference between consecutive terms (called the common difference) is constant. In a geometric series, the ratio of consecutive terms (called the common ratio) is constant.
For the given sequence an, we calculate the first few terms to identify a pattern:
- a1 = 4 - 1/2 = 3.5
- a2 = 4 - 2/2 = 3
- a3 = 4 - 3/2 = 2.5
The difference between consecutive terms is:
- a2 - a1 = 3 - 3.5 = -0.5
- a3 - a2 = 2.5 - 3 = -0.5
Since the difference is constant, we conclude that the series is arithmetic with a common difference of -0.5.