in a geometric sequence, $a_4 = 54$ and $a_7 = 1,458$. what is the 12th term?

5,616
354,294
14,350,365
234,812,358

Question

in a geometric sequence, $a_4 = 54$ and $a_7 = 1,458$. what is the 12th term?

5,616
354,294
14,350,365
234,812,358

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Geometric Sequences", "Common Ratio", "Geometric Sequence Explicit Formula" ], "new_concepts": [], "current_concepts": [ "Geometric Sequences", "Common Ratio", "Geometric Sequence Explicit Formula" ] } ## Find the common ratio Using the **Common Ratio** and **Geometric Sequences** knowledge points $$ \begin{aligned} a_7 &= a_4 \cdot r^{7-4} \ 1458 &= 54 \cdot r^3 \ r^3 &= \frac{1458}{54} = 27 \ r &= 3 \end{aligned} $$ ## Calculate the 12th term Using the **Geometric Sequence Explicit Formula** knowledge point $$ \begin{aligned} a_{12} &= a_7 \cdot r^{12-7} \ a_{12} &= 1458 \cdot 3^5 \ a_{12} &= 1458 \cdot 243 \ a_{12} &= 354294 \end{aligned} $$

5,616

354,294

14,350,365

234,812,358

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Geometric Sequences" ] }

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Question

Explanation:

Find the common ratio

Calculate the 12th term

Answer: