Find the sum of the geometric series 1-3+3^(2)-3^(3)+dots-3^(29) Choose 1 answer: (A) -1.03*10^(14) (B) -6.86*10^(13) (c) -5.15*10^(13) (D) 1.72*10^(13)

Question

Find the sum of the geometric series  1-3+3^(2)-3^(3)+dots-3^(29) Choose 1 answer: (A)  -1.03*10^(14) (B)  -6.86*10^(13) (c)  -5.15*10^(13) (D)  1.72*10^(13)

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Identify Geometric Series: The given series is a geometric series with the first term a = 1 and the common ratio r = -3. The sum of a finite geometric series can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where n is the number of terms.Determine Number of Terms: First, we need to determine the number of terms in the series. Since the series starts at 3^0 (which is 1) and goes up to 3^29, and the powers of 3 increase by 1 each time, there are 30 terms in total.Apply Sum Formula: Now we can apply the formula for the sum of a geometric series: S_n = a(1 - r^n) / (1 - r). Here, a = 1, r = -3, and n = 30.Calculate Numerator: Plugging the values into the formula, we get S_30 = 1(1 - (-3)^30) / (1 - (-3)).Calculate Denominator: Calculating the numerator: 1 - (-3)^30 = 1 - 3^30. Since 3^30 is a very large number, we can leave it in the exponential form for now.Substitute Values: Calculating the denominator: 1 - (-3) = 1 + 3 = 4.Calculate Exponential Value: Now we have S_30 = (1 - 3^30) / 4. To find the exact value of 3^30, we can use a calculator or a computer.Perform Subtraction: Using a calculator, we find that 3^30 is approximately 2.05 * 10^14.Divide by 4: Substituting this value into our sum expression, we get S_30 = (1 - 2.05 * 10^14) / 4.Express in Scientific Notation: Now we perform the subtraction in the numerator: 1 - 2.05 * 10^14 = -2.05 * 10^14 + 1. Since 1 is negligible compared to 2.05 * 10^14, we can approximate this as -2.05 * 10^14.Finally, we divide by 4: S_30 = -2.05 * 10^14 / 4 = -0.5125 * 10^14.We can write -0.5125 * 10^14 in scientific notation as -5.125 * 10^13.

Find the sum of the geometric series $\newline$ $1-3+3^{2}-3^{3}+\ldots-3^{29}$ $\newline$ Choose $1$ answer: $\newline$ (A) $-1.03 \cdot 10^{14}$ $\newline$ (B) $-6.86 \cdot 10^{13}$ $\newline$ (C) $-5.15 \cdot 10^{13}$ $\newline$ (D) $1.72 \cdot 10^{13}$

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