which recursive formula describes the pattern in the perimeters of the images, where $a_n$ is the perimeter of the $n$th image?

$n = 1$
$n = 2$
$n = 3$

$a_n = a_{n-1} + 2, a_1 = 1$
$a_n = a_{n-1} + 1, a_1 = 1$
$a_n = a_{n-1} + 2, a_1 = 4$
$a_n = 2a_{n-1} + 1, a_1 = 4$

Question

which recursive formula describes the pattern in the perimeters of the images, where $a_n$ is the perimeter of the $n$th image?

$n = 1$
$n = 2$
$n = 3$

$a_n = a_{n-1} + 2, a_1 = 1$
$a_n = a_{n-1} + 1, a_1 = 1$
$a_n = a_{n-1} + 2, a_1 = 4$
$a_n = 2a_{n-1} + 1, a_1 = 4$

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", "Arithmetic Sequences" ], "new_concepts": [], "current_concepts": [ "Recursive Formulas", "Arithmetic Sequences", "Sequence Modeling" ] } ## Determine the perimeters of the first three figures $$ \begin{aligned} &a_1 = 4 \quad (\text{square of side length } 1) \ &a_2 = 6 \quad (\text{rectangle of dimensions } 2 \times 1) \ &a_3 = 8 \quad (\text{rectangle of dimensions } 3 \times 1) \end{aligned} $$ ## Find the common difference and recursive relation $$ \begin{aligned} &d = a_2 - a_1 = 6 - 4 = 2 \ &d = a_3 - a_2 = 8 - 6 = 2 \ &a_n = a_{n-1} + 2, \quad a_1 = 4 \end{aligned} $$

$a_n = a_{n-1} + 2, a_1 = 1$

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

$a_n = a_{n-1} + 2, a_1 = 4$

$a_n = 2a_{n-1} + 1, a_1 = 4$

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

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QUESTION IMAGE

Question

Explanation:

Determine the perimeters of the first three figures

Find the common difference and recursive relation

Answer: