The common ratio in a geometric series is 4 and the first term is 3 . Find the sum of the first 8 terms in the series.

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Identify Formula: Identify the formula for the sum of the first n terms in a geometric series. The sum of the first n terms in a geometric series can be found using the formula: 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.Plug Values: Plug the given values into the formula. We are given the first term a = 3, the common ratio r = 4, and the number of terms n = 8. So we substitute these values into the formula: S_8 = 3 * (1 - 4^8) / (1 - 4)Calculate Power: Calculate the power of the common ratio. Calculate 4^8 to simplify the formula. 4^8 = 65536Substitute Value: Substitute the value of 4^8 back into the formula. S_8 = 3 * (1 - 65536) / (1 - 4)Simplify Numerator: Simplify the numerator of the formula. Calculate 1 - 65536. 1 - 65536 = -65535Simplify Denominator: Simplify the denominator of the formula. Calculate 1 - 4. 1 - 4 = -3Divide to Find Sum: Divide the numerator by the denominator to find the sum of the first 8 terms. S_8 = 3 * (-65535) / (-3)Final Answer: Simplify the expression to find the final answer. S_8 = 3 * 65535 / 3 S_8 = 65535

The common ratio in a geometric series is $4$ and the first term is $3$ . $\newline$ Find the sum of the first $8$ terms in the series.

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The common ratio in a geometric series is $4$ and the first term is $3$ . $\newline$ Find the sum of the first $8$ terms in the series.