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 consecutive terms: \$5 - 10 = -5$, \(\frac{5}{2} - 5 = \frac{5}{2} - \frac{10}{2} = -\frac{5}{2}\), and \(\frac{5}{4} - \frac{5}{2} = \frac{5}{4} - \frac{10}{4} = -\frac{5}{4}\). Since these differences are not the same, the sequence is not arithmetic.
3. Next, calculate the ratio between consecutive terms: \(\frac{5}{10} = \frac{1}{2}\), \(\frac{\frac{5}{2}}{5} = \frac{5/2}{5} = \frac{1}{2}\), and \(\frac{\frac{5}{4}}{\frac{5}{2}} = \frac{5/4}{5/2} = \frac{1}{2}\). Since the ratio is constant, the sequence is geometric.
4. Conclude that because the ratio between terms is constant and equal to \(\frac{1}{2}\), the sequence is geometric.
5. Remember, identifying the pattern involves checking both differences and ratios to determine the type of sequence.