Which formula represents a geometric sequence?

The formula that represents a geometric sequence is indeed the expression where ( a_n = a_1 \times r^{(n-1)} ). In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).

This formula captures that relationship nicely: ( a_1 ) is the first term, and by multiplying it by ( r ), raised to the power of ( n-1 ), we obtain the ( n )-th term. The exponent ( n-1 ) reflects how many times we multiply ( a_1 ) by the common ratio to get to the desired term, reinforcing the concept of exponential growth or decay that is characteristic of geometric sequences.

In contrast, other choices represent different types of sequences. The option representing an arithmetic sequence, which adds a constant value ( d ) to the previous term, demonstrates how sequence behavior varies based on whether the operations involved are multiplicative (as in geometric sequences) or additive (as in arithmetic sequences). The formula involving ( r^{(n-1)} ) without a leading term or operation before it does not account for an initial term and thus also does not represent