What is the general term a_n of a geometric sequence given the first term a1 and common ratio r?

• A. a_n = a1 * r^(n-1)
• B. a_n = a1 * r^n
• C. a_n = a1 + r^(n-1)
• D. a_n = a1 * n * r

Answer: (A)

In a geometric sequence, each term is found by multiplying the previous term by the common ratio. Starting from the first term a1, after n−1 multiplications by r you get a_n = a1 · r^(n−1). This aligns with the pattern a2 = a1·r, a3 = a1·r^2, and so on, and it also checks for n = 1 since a1 · r^0 = a1. For example, with a1 = 3 and r = 2, the terms are 3, 6, 12, 24, … and a3 = 3 · 2^2 = 12. The other forms don’t match the way terms scale in a geometric sequence: using a1 · r^n would make a2 = a1 · r^2, which is too large; using addition or a linear n factor doesn’t produce the exponential growth that defines a geometric sequence.