Decision Mathematics 2: Recurrence relationsEdexcel A-Level Further Maths: Topic test

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

   (a)

   What is the value of $u_3$?

   [1 mark]

   • A$10$
   • B$30$
   • C$34$
   • D$22$

   (b)

   What is the constant particular solution of the recurrence relation?

   [1 mark]

   • A$-2$
   • B$2$
   • C$-4$
   • D$\frac43$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sequence satisfies $u_{n+2}-7u_{n+1}+10u_n=0$ with $u_1=7$ and $u_2=29$.

   (a)

   What are the roots of the auxiliary equation?

   [1 mark]

   • A$-2$ and $-5$
   • B$7$ and $10$
   • C$1$ and $10$
   • D$2$ and $5$

   (b)

   Which expression is the general solution of the recurrence relation?

   [1 mark]

   • A$u_n=A\times(-2)^n+B\times(-5)^n$
   • B$u_n=A\times2^n+B\times5^n$
   • C$u_n=(A+Bn)\times2^n$
   • D$u_n=A\times7^n+B\times10^n$

   (c)

   Find the values of $A$ and $B$ in the general solution and state $u_n$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A reservoir holds $8000\ \mathrm{m}^3$ of water at the start of day $1$. Each day $6\%$ of the water in the reservoir evaporates and $300\ \mathrm{m}^3$ of water is pumped in. The volume at the start of day $n$ is $u_n\ \mathrm{m}^3$, where $u_{n+1}=0.94u_n+300$.

   (a)

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

   [3 marks]

   (b)

   Find the first day on which the volume at the start of the day is below $5100\ \mathrm{m}^3$, and describe what happens to the volume in the long term.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number of visitors to a museum in week $n$, in hundreds, is $u_n$, where $u_{n+2}-6u_{n+1}+8u_n=6$, $u_1=12$ and $u_2=30$.

   (a)

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

   [6 marks]

   (b)

   Let $w_n=u_{n+1}-2u_n$. Show that $w_{n+1}=4w_n+6$, and hence solve this first order recurrence relation to find $w_n$ in terms of $n$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A sequence satisfies $u_{n+1}+2u_n=9$ with $u_1=5$.

   (a)

   Which expression is the complementary function?

   [1 mark]

   • A$A\times(-2)^n$
   • B$A\times2^n$
   • C$A\times9^n$
   • D$A\times\left(-\frac12\right)^n$

   (b)

   What is the value of $u_4$?

   [1 mark]

   • A$13$
   • B$11$
   • C$-13$
   • D$103$

   (c)

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

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A sequence satisfies $u_{n+2}-3u_{n+1}+2u_n=4$.

   (a)

   What are the roots of the auxiliary equation?

   [1 mark]

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

   (b)

   Which form should be tried for a particular solution?

   [1 mark]

   • A$\lambda$
   • B$\lambda n^2$
   • C$\lambda\times2^n$
   • D$\lambda n$

   (c)

   Find a particular solution of the recurrence relation.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The temperature deviation $u_n$, in $^\circ\mathrm{C}$, of a chemical vat from its target at the $n$th hourly reading satisfies $u_{n+2}+4u_{n+1}+4u_n=0$, with $u_1=-6$ and $u_2=20$.

   (a)

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

   [3 marks]

   (b)

   Find the first reading for which the deviation has magnitude greater than $1000\,^\circ\mathrm{C}$, and state whether that deviation is above or below the target.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A sequence $u_1, u_2, u_3, \dots$ is defined by the recurrence relation $3u_{n+2}+10u_{n+1}-8u_n=15$, with $u_1=17$ and $u_2=31$.

   (a)

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

   [6 marks]

   (b)

   Let $t_n=u_n-3$. Show that $3t_{n+2}+10t_{n+1}-8t_n=0$. Hence, using your answer to (a), find the least $n$ for which $|u_n-3|>1000$.

   [6 marks]

   Total for question 8: 12 marks

End of questions