sebastian writes the recursive formula $f(x+1) = 4f(x)$ to represent a geometric sequence whose second term is 12. which explicit formula can be used to model the same sequence?

$f(x) = 12(4)^x$
$f(x) = 3(4)^{x-1}$
$f(x) = 4(12)^x$
$f(x) = 4(3)^{x-1}$

Question

sebastian writes the recursive formula $f(x+1) = 4f(x)$ to represent a geometric sequence whose second term is 12. which explicit formula can be used to model the same sequence?

$f(x) = 12(4)^x$
$f(x) = 3(4)^{x-1}$
$f(x) = 4(12)^x$
$f(x) = 4(3)^{x-1}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Geometric Sequences", "Recursive Formulas" ], "new_concepts": [ "Explicit Formula for Geometric Sequence" ], "current_concepts": [ "Geometric Sequences", "Recursive Formulas", "Explicit Formula for Geometric Sequence" ] } ## Find the first term of the sequence $$ \begin{aligned} &f(x+1) = 4f(x) \implies r = 4\ &f(2) = 12\ &f(2) = 4f(1) \implies 12 = 4f(1) \implies f(1) = 3 \end{aligned} $$ ## Determine the explicit formula $$ \begin{aligned} &f(x) = f(1) \cdot r^{x-1}\ &f(x) = 3(4)^{x-1} \end{aligned} $$

(A) $f(x) = 12(4)^x$

(B) $f(x) = 3(4)^{x-1}$

(C) $f(x) = 4(12)^x$

(D) $f(x) = 4(3)^{x-1}$

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

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Explanation:

Find the first term of the sequence

Determine the explicit formula

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