Suppose a city with population 100,000 has been growing at a rate of \( 6 \% \) per year. If this rate continues, find the population of this city in 25 years. The population in 25 years will be approximately \( \square \). (Round to the nearest whole number as needed.) \( 26=11: 59 \mathrm{pm} \) Thes

Given: - Initial population of the city = 100,000 - Annual growth rate = 6% We need to find the population of the city in 25 years. To find the population in 25 years, we can use the formula for compound interest: \[ A = P(1 + r)^n \] where: - \( A \) is the final amount (population in 25 years) - \( P \) is the initial amount (initial population of 100,000) - \( r \) is the annual growth rate (6% or 0.06) - \( n \) is the number of years (25) Substitute the given values into the formula: \[ A = 100,000(1 + 0.06)^{25} \] Now, we can calculate the population in 25 years. Calculate the value by following steps: - step0: Calculate: \(100000\left(1+0.06\right)^{25}\) - step1: Add the numbers: \(100000\times 1.06^{25}\) - step2: Convert the expressions: \(100000\left(\frac{53}{50}\right)^{25}\) - step3: Simplify: \(100000\times \frac{53^{25}}{50^{25}}\) - step4: Rewrite the expression: \(2500\times 40\times \frac{53^{25}}{50^{25}}\) - step5: Rewrite the expression: \(50^{2}\times 40\times \frac{53^{25}}{50^{25}}\) - step6: Reduce the numbers: \(40\times \frac{53^{25}}{50^{23}}\) - step7: Rewrite the expression: \(10\times 4\times \frac{53^{25}}{50^{23}}\) - step8: Rewrite the expression: \(10\times 4\times \frac{53^{25}}{10^{23}\times 5^{23}}\) - step9: Reduce the numbers: \(4\times \frac{53^{25}}{10^{22}\times 5^{23}}\) - step10: Rewrite the expression: \(4\times \frac{53^{25}}{2^{22}\times 5^{22}\times 5^{23}}\) - step11: Rewrite the expression: \(2^{2}\times \frac{53^{25}}{2^{22}\times 5^{22}\times 5^{23}}\) - step12: Reduce the numbers: \(1\times \frac{53^{25}}{2^{20}\times 5^{22}\times 5^{23}}\) - step13: Multiply the fractions: \(\frac{53^{25}}{2^{20}\times 5^{45}}\) The population of the city in 25 years will be approximately 429,187.