What is a key characteristic of geometric sequences?

What is a key characteristic of geometric sequences?

• A. Common difference
• B. Common ratio
• C. Linear progression
• D. Constant term

Answer: (B)

In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. This characteristic is what distinguishes geometric sequences from arithmetic sequences, where a common difference is used. For example, in the geometric sequence 2, 6, 18, 54, the common ratio is 3, as each term is obtained by multiplying the previous term by 3. This means that the relationship between the terms is multiplicative rather than additive. Understanding the concept of a common ratio is crucial in identifying and working with geometric sequences, as it allows one to generalize the n-th term of the sequence using the formula \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio. The other choices reflect properties related to different types of sequences or configurations that are not applicable to geometric sequences. For example, a common difference pertains to arithmetic sequences, linear progression refers to sequences with a constant difference in their terms, and a constant term does not capture the defining feature of how terms in a geometric sequence are generated.

In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. This characteristic is what distinguishes geometric sequences from arithmetic sequences, where a common difference is used.

For example, in the geometric sequence 2, 6, 18, 54, the common ratio is 3, as each term is obtained by multiplying the previous term by 3. This means that the relationship between the terms is multiplicative rather than additive.

Understanding the concept of a common ratio is crucial in identifying and working with geometric sequences, as it allows one to generalize the n-th term of the sequence using the formula ( a_n = a_1 \cdot r^{(n-1)} ), where ( a_1 ) is the first term and ( r ) is the common ratio.

The other choices reflect properties related to different types of sequences or configurations that are not applicable to geometric sequences. For example, a common difference pertains to arithmetic sequences, linear progression refers to sequences with a constant difference in their terms, and a constant term does not capture the defining feature of how terms in a geometric sequence are generated.