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

1. 1
   A sequence satisfies $u_{n+2}-7u_{n+1}+12u_n=0$ for $n\geq1$, with $u_1=2$ and $u_2=2$.

   (a)

   Which of the following gives the roots of the auxiliary equation?

   [1 mark]

   • A$m=-3$ and $m=-4$
   • B$m=2$ and $m=6$
   • C$m=3$ and $m=4$
   • D$m=7$ and $m=12$

   (b)

   Which of the following is the general solution of the recurrence relation?

   [1 mark]

   • A$A\times3^n+B\times4^n$
   • B$(A+Bn)3^n$
   • C$A(-3)^n+B(-4)^n$
   • D$A\times7^n+B\times12^n$

   (c)

   Find $u_n$ in terms of $n$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sequence satisfies $u_{n+2}-6u_{n+1}+9u_n=16$ for $n\geq1$, with $u_1=10$ and $u_2=31$.

   (a)

   Which of the following is the complementary function?

   [1 mark]

   • A$A\times3^n+B\times3^n$
   • B$(A+Bn)3^n$
   • C$A\times3^n+B(-3)^n$
   • D$A\times9^n+B$

   (b)

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

   [1 mark]

   • A$16$
   • B$-4$
   • C$1$
   • D$4$

   (c)

   Find $u_n$ in terms of $n$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence satisfies $u_{n+2}-5u_{n+1}+6u_n=2n+1$ for $n\geq1$, with $u_1=8$ and $u_2=17$.

   (a)

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

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A sequence satisfies $u_{n+2}=5u_{n+1}-6u_n$ for $n\geq1$, with $u_1=1$ and $u_2=5$.

   (a)

   (i) Find $u_n$ in terms of $n$.
   (ii) Hence find the limit of $\frac{u_{n+1}}{u_n}$ as $n\to\infty$.

   [6 marks]

   (b)

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

   [6 marks]

   Total for question 4: 12 marks

End of questions