Find the sum of the following series. Round to the nearest hundredth if necessary. 5+15+45+dots+7971615 Sum of a finite geometric series: S_(n)=(a_(1)-a_(1)r^(n))/(1-r)

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Find the sum of the following series. Round to the nearest hundredth if necessary.  5+15+45+dots+7971615 Sum of a finite geometric series:  S_(n)=(a_(1)-a_(1)r^(n))/(1-r)

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Find Number of Terms: Next, we need to find the number of terms (n) in the series. We can do this by using the formula for the nth term of a geometric series, which is a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, and r is the common ratio. We will solve for n using the last term given, which is 7971615.  7971615 = 5 * 3^(n-1)Isolate n in Equation: To solve for n, we need to isolate n on one side of the equation. We start by dividing both sides by 5.  7971615 / 5 = 3^(n-1) 1594323 = 3^(n-1)Take Logarithm: Now, we take the logarithm of both sides to solve for n. We can use the natural logarithm or the common logarithm; here, we will use the natural logarithm for convenience.  ln(1594323) = ln(3^(n-1))Rewrite Equation: Using the property of logarithms that ln(a^b) = b * ln(a), we can rewrite the equation as:  ln(1594323) = (n-1) * ln(3)Solve for n-1: Now, we divide both sides by ln(3) to solve for n-1.  (n-1) = ln(1594323) / ln(3)Calculate n-1: We calculate the value of n-1 using a calculator.  n-1 = ln(1594323) / ln(3) ≈ 14Find n: To find n, we add 1 to the result.  n = 14 + 1 n = 15Use Sum Formula: Now that we have the number of terms, we can use the formula for the sum of a finite geometric series to find the sum.  S_n = (a_1 - a_1 * r^n) / (1 - r)Substitute Values: We substitute the values we know into the formula.  S_15 = (5 - 5 * 3^15) / (1 - 3)Calculate Numerator and Denominator: We calculate the numerator and the denominator separately.  Numerator: 5 - 5 * 3^15 Denominator: 1 - 3Divide Numerator by Denominator: We calculate the numerator and the denominator using a calculator.  Numerator: 5 - 5 * 3^15 ≈ 5 - 5 * 14348907 ≈ 5 - 71744535 Denominator: 1 - 3 = -2Calculate Final Sum: Now we divide the numerator by the denominator to find the sum.  S_15 = (5 - 71744535) / -2We calculate the final sum using a calculator.  S_15 = (-71744530) / -2 S_15 = 35872265

Find the sum of the following series. Round to the nearest hundredth if necessary. $\newline$ $5+15+45+\ldots+7971615$ $\newline$ Sum of a finite geometric series: $\newline$ $S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r}$

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Find the sum of the following series. Round to the nearest hundredth if necessary.\newline5+15+45+…+7971615 5+15+45+\ldots+7971615 5+15+45+…+7971615\newlineSum of a finite geometric series:\newlineSn=a1−a1rn1−r S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r} Sn​=1−ra1​−a1​rn​

Find the sum of the following series. Round to the nearest hundredth if necessary. $\newline$ $5+15+45+\ldots+7971615$ $\newline$ Sum of a finite geometric series: $\newline$ $S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r}$