### Geometric Sequences: Sum Calculation#### Problem StatementFind the sum of the indicated number of terms of the geometric sequence.\[ -4, 12, -36, \ldots ; \quad n = 4 \](Note: The fourth term is not explicitly given and needs to be calculated.)#### ExplanationThe problem involves finding the sum of the first four terms of a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.Given:- First term (\(a_1\)) = \(-4\)- Common ratio (\(r\)) can be found using the second term divided by the first term: \( r = \frac{12}{-4} = -3 \)- Number of terms (\(n\)) = 4#### Steps to Calculate the SumTo find the sum \(S_n\) of the first \(n\) terms of a geometric sequence, you can use the formula:\[ S_n = a_1 \frac{1-r^n}{1-r} \]For the given sequence:1. \(a_1 = -4\)2. \( r = -3 \)3. \( n = 4 \)First, calculate \(r^n\):\[ r^n = (-3)^4 = 81 \]Applying the sum formula:\[ S_4 = -4 \frac{1 - (-3)^4}{1 - (-3)} = -4 \frac{1 - 81}{1 + 3} = -4 \frac{1 - 81}{4} = -4 \frac{-80}{4} = -4 \times (-20) = 80 \]Therefore, the sum of the first four terms is \(80\).#### Final Answer\[ \boxed{80} \]

Name: ### Geometric Sequences: Sum Calculation#### Problem StatementFind th
Uploaded: 2024-03-20T06:38:58.000Z
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Description: ### Geometric Sequences: Sum Calculation#### Problem StatementFind the sum of the indicated number of terms of the geometric sequence.\[ -4, 12, -36, \ldots ; \quad n = 4 \](Note: The fourth term is not explicitly given and needs to be calculated.)#