Question

use the formula \\(s_n = \frac{n(a_1 + a_n)}{2}\\) to evaluate the following sum.

\\\sum_{k=1}^{75} (-7 + (k - 1) \cdot 2)\\

\\(s_{75} =\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Arithmetic Series",
"Sigma Notation",
"Summation Formula"
],
"new_concepts": [],
"current_concepts": [
"Arithmetic Series",
"Sigma Notation",
"Summation Formula"
]
}
</pre_analysis>

<reasoning>

Identify the parameters of the arithmetic series

\[

$$\begin{aligned} &n = 75 \\ &a_1 = -7 + (1 - 1) \cdot 2 = -7 \\ &a_{75} = -7 + (75 - 1) \cdot 2 = -7 + 74 \cdot 2 = 141 \end{aligned}$$

\]

Apply the arithmetic series sum formula

\[

$$\begin{aligned} S_{75} &= \frac{75 \cdot (a_1 + a_{75})}{2} \\ S_{75} &= \frac{75 \cdot (-7 + 141)}{2} \\ S_{75} &= \frac{75 \cdot 134}{2} \\ S_{75} &= 75 \cdot 67 \\ S_{75} &= 5025 \end{aligned}$$

\]
</reasoning>

<answer>
Use the formula \(S_n = \frac{n(a_1 + a_n)}{2}\) to evaluate the following sum.
\[\sum_{k=1}^{75} (-7 + (k - 1) \cdot 2)\]
\(S_{75} =\) <blank>5025</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Arithmetic Series"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Arithmetic Series",
"Sigma Notation",
"Summation Formula"
],
"new_concepts": [],
"current_concepts": [
"Arithmetic Series",
"Sigma Notation",
"Summation Formula"
]
}
</pre_analysis>

<reasoning>

Identify the parameters of the arithmetic series

\[

$$\begin{aligned} &n = 75 \\ &a_1 = -7 + (1 - 1) \cdot 2 = -7 \\ &a_{75} = -7 + (75 - 1) \cdot 2 = -7 + 74 \cdot 2 = 141 \end{aligned}$$

\]

Apply the arithmetic series sum formula

\[

$$\begin{aligned} S_{75} &= \frac{75 \cdot (a_1 + a_{75})}{2} \\ S_{75} &= \frac{75 \cdot (-7 + 141)}{2} \\ S_{75} &= \frac{75 \cdot 134}{2} \\ S_{75} &= 75 \cdot 67 \\ S_{75} &= 5025 \end{aligned}$$

\]
</reasoning>

<answer>
Use the formula \(S_n = \frac{n(a_1 + a_n)}{2}\) to evaluate the following sum.
\[\sum_{k=1}^{75} (-7 + (k - 1) \cdot 2)\]
\(S_{75} =\) <blank>5025</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Arithmetic Series"
]
}
</post_analysis>