Find the sum of the first 8 terms of the following series, to the nearest integer. 3,(15)/(4),(75)/(16),dots Answer:

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Find the sum of the first 8 terms of the following series, to the nearest integer.  3,(15)/(4),(75)/(16),dots Answer:

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Identify Pattern: Identify the pattern of the series. The series is geometric because each term is multiplied by a common ratio to get the next term. To find the common ratio (r), divide the second term by the first term. $ r = \frac{15/4}{3} = \frac{15}{4} \cdot \frac{1}{3} = \frac{15}{12} = \frac{5}{4} $Use Formula: Use the formula for the sum of the first n terms of a geometric series, which is $ 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. Here, $ a_1 = 3 $, $ r = \frac{5}{4} $, and $ n = 8 $.Calculate Sum: Calculate the sum of the first 8 terms using the formula. $ S_8 = \frac{3(1 - (\frac{5}{4})^8)}{1 - \frac{5}{4}} $Simplify Denominator: Simplify the denominator of the fraction. $ 1 - \frac{5}{4} = \frac{4}{4} - \frac{5}{4} = -\frac{1}{4} $Calculate Exponent: Calculate $ (\frac{5}{4})^8 $ to find the numerator. $ (\frac{5}{4})^8 $ is a large number, so it's best to use a calculator for this step.Substitute and Calculate: Substitute the values into the sum formula and calculate. $ S_8 = \frac{3(1 - (\frac{5}{4})^8)}{-\frac{1}{4}} $ Using a calculator, $ (\frac{5}{4})^8 \approx 1525.87890625 $ $ S_8 = \frac{3(1 - 1525.87890625)}{-\frac{1}{4}} $Simplify Numerator: Simplify the numerator. $ 1 - 1525.87890625 \approx -1524.87890625 $ $ S_8 = \frac{3(-1524.87890625)}{-\frac{1}{4}} $Multiply and Divide: Multiply the numerator by 3 and divide by the denominator. $ S_8 = \frac{-4574.63671875}{-\frac{1}{4}} $ $ S_8 = -4574.63671875 \cdot (-4) $ $ S_8 = 1830.6546875 $Round to Nearest: Round the sum to the nearest integer. $ S_8 \approx 1831 $

Find the sum of the first $8$ terms of the following series, to the nearest integer. $\newline$ $3, \frac{15}{4}, \frac{75}{16}, \ldots$ $\newline$ Answer:

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Find the sum of the first $8$ terms of the following series, to the nearest integer. $\newline$ $3, \frac{15}{4}, \frac{75}{16}, \ldots$ $\newline$ Answer: