\( \sum _ { k = 0 } ^ { 29 } - 3 ( 0.9 ) ^ { k } \approx \)

It seems that the calculation of the sum is too complex and has timed out. Let's try to simplify the expression first and then calculate the sum. The given expression is \( \sum_{k=0}^{29} -3(0.9)^{k} \). We can simplify this expression by using the formula for the sum of a geometric series: \[ \sum_{k=0}^{n} ar^k = \frac{a(1-r^{n+1})}{1-r} \] In this case, \( a = -3 \), \( r = 0.9 \), and \( n = 29 \). Substitute these values into the formula: \[ \sum_{k=0}^{29} -3(0.9)^{k} = \frac{-3(1-0.9^{30})}{1-0.9} \] Now, we can calculate the sum using this simplified expression. Calculate the value by following steps: - step0: Calculate: \(\frac{-3\left(1-0.9^{30}\right)}{1-0.9}\) - step1: Convert the expressions: \(\frac{-3\left(1-\left(\frac{9}{10}\right)^{30}\right)}{1-0.9}\) - step2: Subtract the terms: \(\frac{-3\times \frac{10^{30}-9^{30}}{10^{30}}}{1-0.9}\) - step3: Multiply the numbers: \(\frac{\frac{-3\times 10^{30}+3^{61}}{10^{30}}}{1-0.9}\) - step4: Subtract the numbers: \(\frac{\frac{-3\times 10^{30}+3^{61}}{10^{30}}}{0.1}\) - step5: Convert the expressions: \(\frac{\frac{-3\times 10^{30}+3^{61}}{10^{30}}}{\frac{1}{10}}\) - step6: Multiply by the reciprocal: \(\frac{-3\times 10^{30}+3^{61}}{10^{30}}\times 10\) - step7: Rewrite the expression: \(-\frac{3\times 10^{30}-3^{61}}{10^{30}}\times 10\) - step8: Reduce the numbers: \(-\frac{3\times 10^{30}-3^{61}}{10^{29}}\) - step9: Alternative Form: \(\approx -28.728265\) El valor aproximado de la suma es -28.728265.