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

\\sum_{k=1}^{30} (12 + (k - 1) \cdot 6)\

Question

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

\\sum_{k=1}^{30} (12 + (k - 1) \cdot 6)\

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 parameters from the summation $$ \begin{aligned} &n = 30 \ &a_1 = 12 + (1 - 1) \cdot 6 = 12 \ &a_{30} = 12 + (30 - 1) \cdot 6 = 12 + 29 \cdot 6 = 12 + 174 = 186 \end{aligned} $$ ## Apply the arithmetic series formula $$ \begin{aligned} &S_{30} = \frac{30 \cdot (12 + 186)}{2} \ &S_{30} = 15 \cdot 198 \ &S_{30} = 2970 \end{aligned} $$ Use the formula $S_n = \frac{n(a_1 + a_n)}{2}$ to evaluate the following sum. $$\sum_{k=1}^{30} (12 + (k-1) \cdot 6)$$ $S_{30} =$ 2970 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Arithmetic Series" ] }

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Explanation:

Identify parameters from the summation

Apply the arithmetic series formula

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