In a geometric sequence, what is a3 in terms of a1 and r?

• A. a3 = a1 * r
• B. a3 = a1 * r^3
• C. a3 = a1 * r^2
• D. a3 = a1 + r^2

Answer: (C)

In a geometric sequence, each term is found by multiplying the previous term by the common ratio r, so the nth term is a_n = a_1 * r^{n-1}. For the third term, plug in n = 3: a3 = a1 * r^{3-1} = a1 * r^2. This shows why the correct form is a1 times r^2. The other forms don’t fit because a3 isn’t obtained by multiplying a1 by r (that would give a2), nor by r^3 (that would give a4), and geometric sequences involve multiplication by r, not addition.