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

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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.
The population of the city in 25 years will be approximately 429,187.

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