First-order recurrence relationsEdexcel A-Level Further Maths: Mind map

Exam questions on First-order recurrence relations

1. A pond is stocked with $400$ fish at the start of year $1$. Each year the number of fish increases by $10\%$ and then $30$ more fish are added. Let $u_n$ be the number of fish at the start of year $n$, so that $u_{n+1}=1.1u_n+30$ with $u_1=400$.
   Find the particular solution of the form $u_n=\lambda$.2 marks
2. A loan of $\pounds5000$ is repaid monthly. Each month $0.5\%$ interest is added to the amount owed and then a repayment of $\pounds200$ is made. Let $u_n$ be the amount owed, in pounds, after $n$ repayments, so $u_0=5000$.
   Hence solve the recurrence relation to find $u_n$ in terms of $n$.2 marks
3. The number of bacteria in a culture, in thousands, at the start of day $n$ is $u_n$. Each day the number quadruples and then $9$ thousand bacteria are removed for testing, so $u_{n+1}=4u_n-9$ with $u_1=5$.
   Solve the recurrence relation to find $u_n$ in terms of $n$.3 marks