When is a sequence $\left\{a_n\right\}$ geometric?

• A. When the terms satisfy $a_{n+1}={a_n}^2$ for all $n$.
• B. When the sequence has a constant sum of consecutive terms, meaning $a_n+a_{n+1}=c$ for some constant $c$.
• C. When there is a constant difference $d$ such that $a_{n+1}=a_n+d$ for all $n$.
• D. When there is a constant ratio $r$ such that $a_{n+1}=r{a}_n$ for all $n$ in the domain.

Answer: (D)

Verified step by step guidance

1. Understand that a sequence \( \{a_n\} \) is called geometric if each term after the first is obtained by multiplying the previous term by a fixed constant ratio.
2. This means there exists a constant \( r \) such that for every term in the sequence, the relationship \( a_{n+1} = r \cdot a_n \) holds true for all \( n \).
3. Recognize that this constant \( r \) is called the common ratio of the geometric sequence.
4. Note that this definition distinguishes geometric sequences from arithmetic sequences, where the difference between terms is constant, and from other types of sequences where sums or powers might be constant.
5. Therefore, to verify if a sequence is geometric, check if the ratio \( \frac{a_{n+1}}{a_n} \) is the same for all consecutive terms.