The sum of the first 12 terms of a geometric sequence with 10 as the first term and a common ratio of 0.3 is a.) 14.2857 b.) 7.6923 c.) 32.8719 d.) 7.0001

To find the sum of the first 12 terms of a geometric sequence, we can use the formula for the sum of a geometric series: \[ S_n = \frac{a(1 - r^n)}{1 - r} \] where: - \( S_n \) is the sum of the first n terms - \( a \) is the first term - \( r \) is the common ratio - \( n \) is the number of terms Given: - First term (\( a \)) = 10 - Common ratio (\( r \)) = 0.3 - Number of terms (\( n \)) = 12 Substitute the given values into the formula: \[ S_{12} = \frac{10(1 - 0.3^{12})}{1 - 0.3} \] Now, we can calculate the sum of the first 12 terms of the geometric sequence. Calculate the value by following steps: - step0: Calculate: \(\frac{10\left(1-0.3^{12}\right)}{\left(1-0.3\right)}\) - step1: Remove the parentheses: \(\frac{10\left(1-0.3^{12}\right)}{1-0.3}\) - step2: Convert the expressions: \(\frac{10\left(1-\left(\frac{3}{10}\right)^{12}\right)}{1-0.3}\) - step3: Subtract the numbers: \(\frac{10\times \frac{10^{12}-3^{12}}{10^{12}}}{1-0.3}\) - step4: Multiply the numbers: \(\frac{\frac{10^{12}-3^{12}}{10^{11}}}{1-0.3}\) - step5: Subtract the numbers: \(\frac{\frac{10^{12}-3^{12}}{10^{11}}}{0.7}\) - step6: Convert the expressions: \(\frac{\frac{10^{12}-3^{12}}{10^{11}}}{\frac{7}{10}}\) - step7: Multiply by the reciprocal: \(\frac{10^{12}-3^{12}}{10^{11}}\times \frac{10}{7}\) - step8: Reduce the numbers: \(\frac{10^{12}-3^{12}}{10^{10}}\times \frac{1}{7}\) - step9: Multiply the fractions: \(\frac{10^{12}-3^{12}}{10^{10}\times 7}\) - step10: Multiply: \(\frac{10^{12}-3^{12}}{7\times 10^{10}}\) The sum of the first 12 terms of the geometric sequence is approximately 14.2857. Therefore, the correct answer is a.) 14.2857.