The geometric sequence a_(i) is defined by the formula: {:[a_(1)=-(1)/(3)],[a_(i)=a_(i-1)*(-3)]:} Find the sum of the first 75 terms in the sequence. Choose 1 answer: (A) -1.01*10^(35) (B) -6.76*10^(34) (c) -5.06*10^(34) (D) 1.69*10^(34)

Question

The geometric sequence  a_(i) is defined by the formula:  {:[a_(1)=-(1)/(3)],[a_(i)=a_(i-1)*(-3)]:} Find the sum of the first 75 terms in the sequence. Choose 1 answer: (A)  -1.01*10^(35) (B)  -6.76*10^(34) (c)  -5.06*10^(34) (D)  1.69*10^(34)

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Identify Terms: Identify the first term and the common ratio of the geometric sequence. The first term $ a_1 $ is given as $ -\frac{1}{3} $. The common ratio $ r $ is the factor between consecutive terms, which is given as $ -3 $.Use Sum Formula: Use the formula for the sum of the first $ n $ terms of a geometric sequence. The sum $ S_n $ of the first $ n $ terms of a geometric sequence is given by the formula: $$ S_n = \frac{a_1(1 - r^n)}{1 - r} $$ where $ a_1 $ is the first term, $ r $ is the common ratio, and $ n $ is the number of terms.Plug in Values: Plug in the values for $ a_1 $, $ r $, and $ n $ into the sum formula. Here, $ a_1 = -\frac{1}{3} $, $ r = -3 $, and $ n = 75 $. $$ S_{75} = \frac{-\frac{1}{3}(1 - (-3)^{75})}{1 - (-3)} $$Simplify Denominator: Simplify the denominator. Since $ 1 - (-3) = 1 + 3 = 4 $, the formula becomes: $$ S_{75} = \frac{-\frac{1}{3}(1 - (-3)^{75})}{4} $$Calculate Exponent: Calculate $ (-3)^{75} $. Since $ -3 $ is raised to an odd power, the result will be negative. The magnitude will be very large, so we will not calculate the exact value but understand that it is a very large negative number.Simplify Numerator: Simplify the numerator. The term $ 1 - (-3)^{75} $ will be a very large positive number because $ (-3)^{75} $ is a very large negative number and subtracting it from 1 gives a positive result.Calculate Sum: Calculate the sum $ S_{75} $. Since the numerator is a large positive number and the denominator is 4, the sum $ S_{75} $ will be a large negative number when multiplied by $ -\frac{1}{3} $.Determine Magnitude: Determine the magnitude of the sum. The magnitude of the sum will be on the order of $ 3^{75} $, which is a number much larger than any of the choices given. Therefore, we can eliminate choices (D) and (A) because they are positive and too small, respectively.Choose Correct Answer: Choose the correct answer based on the magnitude. Between choices (B) and (C), choice (B) has the larger magnitude, which aligns with our expectation of a very large negative number. Therefore, the correct answer is (B) $ -6.76 \times 10^{34} $.

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