Population after 'n' years, with a 2r% yearly increase: \(p \times (1 + \frac{2r}{100})^n\) formula describes it.
The population of a village at present is represented by 'p'. If the population increases at a rate of 2r% per year, the population after 'n' years can be calculated using a formula for exponential growth.
The increase rate of 2r% means that the population grows by a factor of (1 + 2r/100) each year. So, after the first year, the population would be p * (1 + 2r/100). After the second year, it would be this value multiplied by (1 + 2r/100) again, resulting in p * (1 + 2r/100)^2. This pattern continues for 'n' years.
Therefore, the population after 'n' years, considering an increase of 2r% annually, is given by the formula:
\[ \text{Population after } n \text{ years} = p \times (1 + \frac{2r}{100})^n \]
This formula uses the initial population 'p', the growth rate of 2r%, and 'n' years to calculate the future population.
For instance, after 5 years, the population would be \( p \times (1 + \frac{2r}{100})^5 \), after 10 years it would be \( p \times (1 + \frac{2r}{100})^{10} \), and so on.
This exponential growth formula accounts for the compounded increase of the population by the specified rate over the given number of years, 'n'.
Question :At present the population of a village is p and if increase rate of population per year be 2r%, the population will be after n years