Identify if each sequence is arithmetic, geometric, or neither.
$10,5,\frac{5}{2},\frac{5}{4},\dots$

• A. Arithmetic
• B. Geometric
• C. Neither

Answer: (B)

Verified step by step guidance

1. First, recall the definitions: an arithmetic sequence has a constant difference between consecutive terms, while a geometric sequence has a constant ratio between consecutive terms.
2. Calculate the difference between the first two terms: \$5 - 10 = -5$. Then check the difference between the next terms: \(\frac{5}{2} - 5 = \frac{5}{2} - \frac{10}{2} = -\frac{5}{2}\). Since these differences are not equal, the sequence is not arithmetic.
3. Next, calculate the ratio between the first two terms: \(\frac{5}{10} = \frac{1}{2}\). Then check the ratio between the next terms: \(\frac{\frac{5}{2}}{5} = \frac{5}{2} \times \frac{1}{5} = \frac{1}{2}\). Since the ratios are equal so far, continue checking the next ratio: \(\frac{\frac{5}{4}}{\frac{5}{2}} = \frac{5}{4} \times \frac{2}{5} = \frac{1}{2}\).
4. Since the ratio between each pair of consecutive terms is constant and equal to \(\frac{1}{2}\), the sequence is geometric.
5. Therefore, identify the sequence as geometric based on the constant ratio found.