Question

choose the function that represents the triangular numbers.

\\(y = 0.5x + 0.5\\)
\\(y = 0.5^x + 0.5x\\)
\\(y = \frac{x}{0.5} + 0.5x\\)
\\(y = 0.5x^2 + 0.5x\\)

Explanation:

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<reasoning>

Identify the values from the given pattern

\[

$$\begin{aligned} &\text{For } x = 2, \quad y = 3 \\ &\text{For } x = 3, \quad y = 6 \\ &\text{For } x = 4, \quad y = 10 \\ &\text{For } x = 5, \quad y = 15 \\ &\text{For } x = 6, \quad y = 21 \end{aligned}$$

\]

Test the given options using \(x = 2\)

\[

$$\begin{aligned} &\text{Option 1: } y = 0.5(2) + 0.5 = 1.5 eq 3 \\ &\text{Option 2: } y = 0.5^2 + 0.5(2) = 0.25 + 1 = 1.25 eq 3 \\ &\text{Option 3: } y = \frac{2}{0.5} + 0.5(2) = 4 + 1 = 5 eq 3 \\ &\text{Option 4: } y = 0.5(2^2) + 0.5(2) = 2 + 1 = 3 \end{aligned}$$

\]

Verify the correct option with \(x = 3\)

\[
y = 0.5(3^2) + 0.5(3) = 4.5 + 1.5 = 6
\]
</reasoning>

<answer>
<mcq-option>\(y = 0.5x + 0.5\)</mcq-option>
<mcq-option>\(y = 0.5^x + 0.5x\)</mcq-option>
<mcq-option>\(y = \frac{x}{0.5} + 0.5x\)</mcq-option>
<mcq-correct>\(y = 0.5x^2 + 0.5x\)</mcq-correct>
</answer>

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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Triangular Numbers"
]
}
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Answer:

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"Triangular Numbers"
],
"new_concepts": [],
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"Triangular Numbers"
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</pre_analysis>

<reasoning>

Identify the values from the given pattern

\[

$$\begin{aligned} &\text{For } x = 2, \quad y = 3 \\ &\text{For } x = 3, \quad y = 6 \\ &\text{For } x = 4, \quad y = 10 \\ &\text{For } x = 5, \quad y = 15 \\ &\text{For } x = 6, \quad y = 21 \end{aligned}$$

\]

Test the given options using \(x = 2\)

\[

$$\begin{aligned} &\text{Option 1: } y = 0.5(2) + 0.5 = 1.5 eq 3 \\ &\text{Option 2: } y = 0.5^2 + 0.5(2) = 0.25 + 1 = 1.25 eq 3 \\ &\text{Option 3: } y = \frac{2}{0.5} + 0.5(2) = 4 + 1 = 5 eq 3 \\ &\text{Option 4: } y = 0.5(2^2) + 0.5(2) = 2 + 1 = 3 \end{aligned}$$

\]

Verify the correct option with \(x = 3\)

\[
y = 0.5(3^2) + 0.5(3) = 4.5 + 1.5 = 6
\]
</reasoning>

<answer>
<mcq-option>\(y = 0.5x + 0.5\)</mcq-option>
<mcq-option>\(y = 0.5^x + 0.5x\)</mcq-option>
<mcq-option>\(y = \frac{x}{0.5} + 0.5x\)</mcq-option>
<mcq-correct>\(y = 0.5x^2 + 0.5x\)</mcq-correct>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Triangular Numbers"
]
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