look at this image.
$x$ 1 2 3 4 5
$f(x)$
what type of function is represented in a pentagonal number pattern?
linear
quadratic
exponential
logarithmic

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Pentagonal Numbers", "Figurate Numbers" ], "new_concepts": [], "current_concepts": [ "Pentagonal Numbers", "Figurate Numbers", "Quadratic Modeling" ] } ## Count the dots for each term $$ \begin{aligned} f(1) &= 1 \ f(2) &= 5 \ f(3) &= 12 \ f(4) &= 22 \ f(5) &= 35 \end{aligned} $$ ## Calculate the first differences $$ \begin{aligned} d_1(1) &= 5 - 1 = 4 \ d_1(2) &= 12 - 5 = 7 \ d_1(3) &= 22 - 12 = 10 \ d_1(4) &= 35 - 22 = 13 \end{aligned} $$ ## Calculate the second differences $$ \begin{aligned} d_2(1) &= 7 - 4 = 3 \ d_2(2) &= 10 - 7 = 3 \ d_2(3) &= 13 - 10 = 3 \end{aligned} $$ Since the second differences are constant, the pattern is represented by a quadratic function.

linear

quadratic

exponential

logarithmic

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Pentagonal Numbers" ] }

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QUESTION IMAGE

Question

Explanation:

Count the dots for each term

Calculate the first differences

Calculate the second differences

Answer: