Final answer:
The sequence 3, 5, 7, ... is arithmetic with a common difference of 2.Therefore, the correct option is a) Arithmetic; 2.
Step-by-step explanation:
To determine whether a sequence is arithmetic or geometric, we need to look at the pattern between consecutive terms. **Arithmetic Sequence:** In an arithmetic sequence, the same number is added or subtracted from one term to get the next term.
This number is known as the common difference, denoted as \( d \). **Geometric Sequence:** In a geometric sequence, each term is multiplied or divided by a constant number to get the next term. This constant number is referred to as the common ratio, denoted as \( r \).
Given the sequence \( 3, 5, 7, \ldots \), we will compute both the differences and ratios between terms to determine the sequence type. **Step 1: Compute the Common Difference** To check for an arithmetic sequence, we find the difference between consecutive terms: Difference between second and first term: \( 5 - 3 = 2 \) Difference between third and second term: \( 7 - 5 = 2 \) Since the differences are the same (\( 2 \)), we have an arithmetic sequence. **Step
2: Compute the Common Ratio** To verify the conclusion, let's also compute the ratios between consecutive terms, which would be relevant if the sequence were geometric: Ratio between second and first term: \( \frac{5}{3} \) Ratio between third and second term: \( \frac{7}{5} \) Because these ratios are not the same, this rules out the possibility of a geometric sequence.
**Conclusion:** The given sequence (3, 5, 7, ...) is an arithmetic sequence with a common difference of \( 2 \). Therefore, the correct answer is **(a) Arithmetic; 2**.
So, the common difference is 2. Therefore, the correct option is a) Arithmetic; 2.