Question

which formula can be used to find the nth term in a geometric sequence where a₁ = 3 and r = 2?
○ aₙ = 3 + 2(n - 1)
○ aₙ = 3(n - 1) + 2
○ aₙ = 3ⁿ⁻¹ · 2
○ aₙ = 3 · 2ⁿ⁻¹

Explanation:

Step1: Recall Geometric Sequence Formula

The formula for the \( n \)-th term of a geometric sequence is \( a_n = a_1 \cdot r^{n - 1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step2: Identify \( a_1 \) and \( r \)

Given \( a_1 = 3 \) and \( r = 2 \) (assuming the typo in the problem, \( r = 2 \) as per the options).

Step3: Substitute into Formula

Substitute \( a_1 = 3 \) and \( r = 2 \) into \( a_n = a_1 \cdot r^{n - 1} \), we get \( a_n = 3 \cdot 2^{n - 1} \).

Step4: Analyze Options

• Option 1: \( a_n = 3 + 2(n - 1) \) is arithmetic sequence formula (linear), not geometric.
• Option 2: \( a_n = 3(n - 1)+2 \) is also linear, not geometric.
• Option 3: Formatting issue, likely typo.
• Option 4: \( a_n = 3 \cdot 2^{n - 1} \) matches the geometric sequence formula with \( a_1 = 3 \) and \( r = 2 \).

Answer:

\( a_n = 3 \cdot 2^{n - 1} \) (the last option, assuming the options are ordered as shown, the one with \( a_n = 3 \cdot 2^{n - 1} \))