Science:Math Exam Resources/Courses/MATH103/April 2016/Question 06 (b) (i)/Solution 1

The population after one year is simply $x_{1}.$ We use the recurrence relation (with $n=0$) and the fact that $x_{0}={\frac {1}{5}}$ to get

$x_{1}={\frac {1}{2}}x_{0}(1-x_{0})+x_{0}={\frac {1}{2}}\cdot {\frac {1}{5}}\left(1-{\frac {1}{5}}\right)+{\frac {1}{5}}={\frac {1}{10}}\cdot {\frac {4}{5}}+{\frac {1}{5}}=\color {blue}{\frac {7}{25}}.$