Find the sum of the first 30 terms in this geometric series: -5-4-(16)/(5)dots Choose 1 answer: (A) -1.24*10^(20) (B) -24.97 (c) -2.78 (D) -0.04

Question

Find the sum of the first 30 terms in this geometric series:  -5-4-(16)/(5)dots Choose 1 answer: (A)  -1.24*10^(20) (B) -24.97 (c) -2.78 (D) -0.04

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Identify first term and common ratio: Identify the first term (a1) and the common ratio (r) of the geometric series. The first term a1 is -5. To find the common ratio r, we divide the second term by the first term. r = -4 / -5 = 4/5Use sum formula for geometric series: Use the formula for the sum of the first n terms of a geometric series: S_n = a1 * (1 - r^n) / (1 - r), where n is the number of terms. Here, n = 30, a1 = -5, and r = 4/5.Plug values and calculate sum: Plug the values into the formula and calculate the sum. S_30 = -5 * (1 - (4/5)^30) / (1 - 4/5)Simplify denominator: Simplify the denominator 1 - 4/5. 1 - 4/5 = 1/5Calculate (4/5)^30: Calculate (4/5)^30. This is a very small number because 4/5 is less than 1 and raising it to the 30th power will make it much smaller.Substitute values into formula: Substitute the values into the formula. S_30 = -5 * (1 - (4/5)^30) / (1/5)Multiply by 5 to simplify: Multiply both numerator and denominator by 5 to simplify the fraction. S_30 = -5 * 5 * (1 - (4/5)^30)Simplify the multiplication: Simplify the multiplication. S_30 = -25 * (1 - (4/5)^30)Approximate sum: Since (4/5)^30 is a very small number, we can approximate 1 - (4/5)^30 as being very close to 1. S_30 ≈ -25 * 1Calculate approximate sum: Calculate the approximate sum. S_30 ≈ -25

Find the sum of the first $30$ terms in this geometric series: $\newline$ $-5-4-\frac{16}{5} \ldots$ $\newline$ Choose $1$ answer: $\newline$ (A) $-1.24 \cdot 10^{20}$ $\newline$ (B) $-24$ . $97$ $\newline$ (C) $-2$ . $78$ $\newline$ (D) $-0$ . $04$

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