a sequence is defined by the explicit formula $a_n = 3^n + 4$. which recursive formula represents the same sequence of numbers?

$a_n = 3a_{n-2} + 4, a_1 = 7$
$a_n = 3n + a_{n-1}, a_1 = 7$
$a_n = 3a_{n-1} - 8, a_1 = 7$
$a_n = n^3 + 6, a_1 = 7$

Question

a sequence is defined by the explicit formula $a_n = 3^n + 4$. which recursive formula represents the same sequence of numbers?

$a_n = 3a_{n-2} + 4, a_1 = 7$
$a_n = 3n + a_{n-1}, a_1 = 7$
$a_n = 3a_{n-1} - 8, a_1 = 7$
$a_n = n^3 + 6, a_1 = 7$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Recursive Formulas", "Sequence Modeling" ], "new_concepts": [], "current_concepts": [ "Recursive Formulas", "Sequence Modeling", "Geometric Sequences" ] } ## Calculate the first few terms of the explicit sequence $$ \begin{aligned} a_n &= 3^n + 4 \ a_1 &= 3^1 + 4 = 7 \ a_2 &= 3^2 + 4 = 13 \ a_3 &= 3^3 + 4 = 31 \end{aligned} $$ ## Express the recursive relationship $$ \begin{aligned} a_n - 4 &= 3^n \ a_{n-1} - 4 &= 3^{n-1} \ a_n - 4 &= 3(a_{n-1} - 4) \ a_n - 4 &= 3a_{n-1} - 12 \ a_n &= 3a_{n-1} - 8 \end{aligned} $$ ## Verify the recursive formula with calculated terms $$ \begin{aligned} a_1 &= 7 \ a_2 &= 3(7) - 8 = 13 \ a_3 &= 3(13) - 8 = 31 \end{aligned} $$

$a_n = 3a_{n-2} + 4, a_1 = 7$

$a_n = 3n + a_{n-1}, a_1 = 7$

$a_n = 3a_{n-1} - 8, a_1 = 7$

$a_n = n^3 + 6, a_1 = 7$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Recursive Formulas" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the first few terms of the explicit sequence

Express the recursive relationship

Verify the recursive formula with calculated terms

Answer: