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

1. 1
   A sequence is defined by $u_{n+1}-5u_n=8$ for $n\geq1$, with $u_1=1$.

   (a)

   Which of the following is the complementary function of the recurrence relation?

   [1 mark]

   • A$A(-5)^n$
   • B$A\times5^n$
   • C$A\times8^n$
   • D$An\times5^n$

   (b)

   Which of the following is the constant particular solution $u_n=\lambda$?

   [1 mark]

   • A$2$
   • B$\frac43$
   • C$8$
   • D$-2$

   (c)

   Hence find $u_n$ in terms of $n$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A fish farm models the number of fish $P_n$ in a lake at the start of year $n$ by $P_{n+1}=1.15P_n-60$, with $P_1=500$. Each year the population grows by 15% and then 60 fish are removed.

   (a)

   Find $P_2$.

   [1 mark]

   • A$515$
   • B$575$
   • C$506$
   • D$635$

   (b)

   Which of the following is the constant particular solution of the recurrence relation?

   [1 mark]

   • A$-400$
   • B$\frac{60}{1.15}\approx52.2$
   • C$400$
   • D$500$

   (c)

   Given that $P_n=100(1.15)^{n-1}+400$, find the first year in which the population exceeds $1000$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence satisfies $u_{n+1}-2u_n=6n$ for $n\geq1$, with $u_1=5$.

   (a)

   Find the complementary function and a particular solution of the form $u_n=an+b$.

   [3 marks]

   (b)

   Hence find $u_n$ in terms of $n$, and find $u_{10}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A sequence is defined by $u_{n+1}=2u_n+3$ for $n\geq1$, with $u_1=5$.

   (a)

   (i) Find $u_n$ in terms of $n$ by solving the recurrence relation.
   (ii) Hence find the smallest $n$ for which $u_n>10^6$.

   [6 marks]

   (b)

   Prove by induction that $u_n=2^{n+2}-3$ for all positive integers $n$.

   [6 marks]

   Total for question 4: 12 marks

End of questions