evaluate:
\\sum_{n=1}^{6} 5(2)^{n-1}\
\s_n = ?\
finite geometric series: \s_n = \frac{a(1-r^n)}{1-r}\

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Geometric Series", "Sigma Notation", "First Term Identification" ], "new_concepts": [], "current_concepts": [ "Geometric Series", "Sigma Notation", "First Term Identification", "Finite Geometric Series Sum" ] } ## Identify series parameters Using the *Sigma Notation* and *First Term Identification* knowledge points $$ \begin{aligned} &\sum_{n=1}^{6} 5(2)^{n-1}\ &a = 5(2)^{1-1} = 5\ &r = 2\ &n = 6 \end{aligned} $$ ## Calculate the finite sum Using the *Geometric Series* knowledge point $$ \begin{aligned} &S_6 = \frac{5(1 - 2^6)}{1 - 2}\ &S_6 = \frac{5(1 - 64)}{-1}\ &S_6 = \frac{5(-63)}{-1} = 315 \end{aligned} $$ Evaluate: $$\sum_{n=1}^{6} 5(2)^{n-1}$$ $S_n =$ 315 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Geometric Series" ] }

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Explanation:

Identify series parameters

Calculate the finite sum

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