Find the sum of the first 45 terms in this geometric series: -0.5+1.5-4.5 dots Choose 1 answer: (A) -7.39*10^(20) (B) -4.92*10^(20) (C) -3.69*10^(20) (D) 1.23*10^(20)

Question

Find the sum of the first 45 terms in this geometric series:  -0.5+1.5-4.5 dots Choose 1 answer: (A)  -7.39*10^(20) (B)  -4.92*10^(20) (C)  -3.69*10^(20) (D)  1.23*10^(20)

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Identify terms and ratio: Identify the first term (a) and the common ratio (r) of the geometric series. The first term a = -0.5, and each term is multiplied by -3 to get the next term (since -0.5 * -3 = 1.5, 1.5 * -3 = -4.5, and so on), so the common ratio r = -3.Use sum formula: Use the formula for the sum of the first n terms of a geometric series: S_n = a(1 - r^n) / (1 - r), where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms. Here, n = 45, a = -0.5, and r = -3.Calculate denominator: Substitute the values into the formula and calculate the sum. S_45 = -0.5(1 - (-3)^45) / (1 - (-3))Calculate exponent: Calculate the denominator of the fraction: 1 - (-3) = 1 + 3 = 4.Estimate numerator: Calculate (-3)^45. Since 45 is an odd number, the result will be negative, and the magnitude will be 3^45.Substitute values: The magnitude of 3^45 is a very large number, and it's not practical to calculate it exactly without a calculator. However, we can estimate that it will be a number much larger than 10^20, and since it's raised to an odd power, the result will be negative.Consider magnitude: Now, calculate the numerator of the fraction: 1 - (-3)^45. Since (-3)^45 is negative and its magnitude is much larger than 1, the result will be approximately equal to -(-3)^45.Eliminate options: Substitute the calculated values into the sum formula: S_45 ≈ -0.5 * (-(-3)^45) / 4 S_45 ≈ 0.5 * 3^45 / 4Choose closest option: Since 3^45 is much larger than 4, dividing by 4 will not significantly affect the magnitude of the number. Therefore, the sum will be approximately 0.5 * 3^45.Given the choices, we are looking for a negative number with a magnitude in the order of 10^20. The only negative options are (A), (B), and (C).Since the sum is approximately 0.5 * 3^45 and we know 3^45 is a very large number, we can eliminate the smallest magnitude among the negative options, which is (C) -3.69*10^20.Between (A) and (B), we choose the one that is closest to half of 3^45. Without an exact calculation, we cannot determine whether (A) -7.39*10^20 or (B) -4.92*10^20 is the correct answer. However, since we are dividing by 4 in the last step, it is more likely that the sum is closer to (B) -4.92*10^20 than to (A) -7.39*10^20.

Find the sum of the first $45$ terms in this geometric series: $\newline$ $-0.5+1.5-4.5 \ldots$ $\newline$ Choose $1$ answer: $\newline$ (A) $-7.39 \cdot 10^{20}$ $\newline$ (B) $-4.92 \cdot 10^{20}$ $\newline$ (C) $-3.69 \cdot 10^{20}$ $\newline$ (D) $1.23 \cdot 10^{20}$

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