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

1. 1
   A sequence satisfies $u_{n+2}=5u_{n+1}-6u_n$ with $u_1=5$ and $u_2=13$.

   (a)

   What are the roots of the auxiliary equation?

   [1 mark]

   • A$1$ and $6$
   • B$-2$ and $-3$
   • C$2$ and $3$
   • D$5$ and $6$

   (b)

   What is the general solution of the recurrence relation?

   [1 mark]

   • A$A\,2^n+B\,3^n$
   • B$A(-2)^n+B(-3)^n$
   • C$(A+Bn)\,2^n$
   • D$A\,6^n$

   (c)

   Find $u_n$ in terms of $n$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sequence satisfies $u_{n+2}=6u_{n+1}-9u_n$ with $u_1=3$ and $u_2=18$.

   (a)

   Which statement describes the roots of the auxiliary equation?

   [1 mark]

   • ATwo distinct roots, $3$ and $-3$
   • BOne repeated root, $3$
   • COne repeated root, $-3$
   • DComplex roots

   (b)

   What is the general solution of the recurrence relation?

   [1 mark]

   • A$A\,3^n+B\,3^n$
   • B$A\,3^n+B(-3)^n$
   • C$A\,3^n+B\,9^n$
   • D$(A+Bn)\,3^n$

   (c)

   Find $u_n$ in terms of $n$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence satisfies $u_{n+2}-u_{n+1}-6u_n=12$ with $u_1=2$ and $u_2=20$.

   (a)

   Find the complementary function and a particular solution of the form $u_n=\lambda$.

   [3 marks]

   (b)

   Hence find $u_n$ in terms of $n$, and find $u_6$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A conservation team models the population of a bird species, in hundreds, at the start of year $n$ by $u_{n+2}=0.9u_{n+1}-0.2u_n+12$, with $u_1=20$ and $u_2=31$.

   (a)

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

   [6 marks]

   (b)

   Describe the long-term behaviour of the population, justify that it never reaches $4000$ birds, and find the first year in which the population exceeds $3900$ birds.

   [6 marks]

   Total for question 4: 12 marks

End of questions