determine the value of the sum below.

\\sum_{k=4}^{10} (-3k + 5)\

Question

determine the value of the sum below.

\\sum_{k=4}^{10} (-3k + 5)\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "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 properties of the arithmetic series $$ \begin{aligned} &a_k = -3k + 5\ &n = 10 - 4 + 1 = 7\ &a_1 = a_4 = -3(4) + 5 = -7\ &a_n = a_{10} = -3(10) + 5 = -25 \end{aligned} $$ ## Apply the arithmetic series sum formula $$ \begin{aligned} &S_n = \frac{n}{2}(a_1 + a_n)\ &S_7 = \frac{7}{2}(-7 + (-25))\ &S_7 = \frac{7}{2}(-32)\ &S_7 = 7 \times (-16) = -112 \end{aligned} $$ $-112$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Arithmetic Series" ] }

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Explanation:

Identify the properties of the arithmetic series

Apply the arithmetic series sum formula

Answer: