First-order recurrence relationsEdexcel A-Level Further Maths: Subtopic test

1. 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 $u_1=400$.

   (a)

   Find $u_2$.

   [1 mark]

   • A$440$
   • B$430$
   • C$473$
   • D$470$

   (b)

   What is the complementary function of $u_{n+1}-1.1u_n=30$?

   [1 mark]

   • A$A(-1.1)^n$
   • B$A(1.1)^n$
   • C$A(30)^n$
   • D$An(1.1)^n$

   (c)

   Find the particular solution of the form $u_n=\lambda$.

   [2 marks]

   Total for question 1: 4 marks

2. 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 $u_0=5000$.

   (a)

   Which recurrence relation models the amount owed?

   [1 mark]

   • A$u_{n+1}=1.005u_n-200$
   • B$u_{n+1}=1.05u_n-200$
   • C$u_{n+1}=1.005(u_n-200)$
   • D$u_{n+1}=0.005u_n-200$

   (b)

   What is the particular solution of $u_{n+1}-1.005u_n=-200$ of the form $u_n=\lambda$?

   [1 mark]

   • A$-40000$
   • B$200$
   • C$40000$
   • D$4000$

   (c)

   Hence solve the recurrence relation to find $u_n$ in terms of $n$.

   [2 marks]

   Total for question 2: 4 marks

3. 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 $u_1=5$.

   (a)

   Solve the recurrence relation to find $u_n$ in terms of $n$.

   [3 marks]

   (b)

   Find the first day on which the culture contains more than one million bacteria.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A drug is taken once each morning. Between consecutive doses $25\%$ of the drug in the body is lost. A dose of $d$ mg is taken each morning, and $u_n$ mg is the amount in the body just after the $n$th dose, so $u_{n+1}=0.75u_n+d$ with $u_1=d$.

   (a)

   A patient takes $d=200$.
   (i) Show that $u_2=350$.
   (ii) Solve the recurrence relation to find $u_n$ in terms of $n$.
   (iii) Find the amount of the drug that the body approaches in the long run.

   [6 marks]

   (b)

   A second patient takes a dose of $d$ mg each morning, and the amount in the body just after each dose approaches $600$ mg in the long run. Find $d$, and find the first dose after which the amount in the body exceeds $590$ mg.

   [6 marks]

   Total for question 4: 12 marks

End of questions