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)\

$s_{75} =$

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)\

$s_{75} =$

Accepted Answer

{ "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" ] } ## 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 $$ \begin{aligned} S_{75} &= \frac{75 \cdot (a_1 + a_{75})}{2} \ S_{75} &= \frac{75 \cdot (-7 + 141)}{2} \ S_{75} &= \frac{75 \cdot 134}{2} \ S_{75} &= 75 \cdot 67 \ S_{75} &= 5025 \end{aligned} $$ Use the formula $S_n = \frac{n(a_1 + a_n)}{2}$ to evaluate the following sum. $$\sum_{k=1}^{75} (-7 + (k - 1) \cdot 2)$$ $S_{75} =$ 5025 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Arithmetic Series" ] }

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Question

Explanation:

Identify the parameters of the arithmetic series

Apply the arithmetic series sum formula

Answer: