evaluate:
\\sum_{n=1}^{4} 8(3)^{n-1}\
\s_n = ?\
finite geometric series: \s_n = \frac{a(1-r^n)}{1-r}\

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Geometric Series", "Sigma Notation", "First Term Identification", "Sum of Finite Geometric Series" ], "new_concepts": [], "current_concepts": [ "Geometric Series", "Sigma Notation", "First Term Identification", "Sum of Finite Geometric Series" ] } ## Identify the parameters of the geometric series $$ \begin{aligned} &\sum_{n=1}^{4} 8(3)^{n-1}\ &a = 8(3)^{1-1} = 8\ &r = 3\ &n = 4 \end{aligned} $$ ## Apply the sum formula for a finite geometric series $$ \begin{aligned} &S_n = \frac{a(1 - r^n)}{1 - r}\ &S_4 = \frac{8(1 - 3^4)}{1 - 3}\ &S_4 = \frac{8(1 - 81)}{-2} \end{aligned} $$ ## Calculate the final sum $$ \begin{aligned} &S_4 = \frac{8(-80)}{-2}\ &S_4 = -4 \times (-80)\ &S_4 = 320 \end{aligned} $$ Evaluate: $$\sum_{n=1}^{4} 8(3)^{n-1}$$ $S_n =$ 320 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Sum of Finite Geometric Series" ] }

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Question

Explanation:

Identify the parameters of the geometric series

Apply the sum formula for a finite geometric series

Calculate the final sum

Answer: