Further Pure 2: Further sequences and seriesEdexcel A-Level Further Maths: Topic test

1. 1
   A sequence satisfies $u_{n+1}-4u_n=9$ for $n\ge1$, with $u_1=2$.

   (a)

   What is the value of $u_3$?

   [1 mark]

   • A$77$
   • B$17$
   • C$68$
   • D$26$

   (b)

   The particular solution is a constant $\lambda$. What is $\lambda$?

   [1 mark]

   • A$3$
   • B$-3$
   • C$\frac95$
   • D$-\frac95$

   (c)

   Find $u_n$ in terms of $n$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sequence satisfies $u_{n+2}-3u_{n+1}-10u_n=0$ for $n\ge1$.

   (a)

   What are the roots of the auxiliary equation?

   [1 mark]

   • A$-5$ and $2$
   • B$3$ and $-10$
   • C$5$ and $-2$
   • D$5$ and $2$

   (b)

   Which expression is the general solution, where $A$ and $B$ are constants?

   [1 mark]

   • A$A\cdot5^n+B\cdot2^n$
   • B$A\cdot(-5)^n+B\cdot2^n$
   • C$(A+Bn)\cdot5^n$
   • D$A\cdot5^n+B\cdot(-2)^n$

   (c)

   Given that $u_1=1$ and $u_2=33$, find the values of $A$ and $B$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence satisfies $u_{n+1}-3u_n=4n$ for $n\ge1$, with $u_1=1$.

   (a)

   Find a particular solution of the form $u_n=\lambda n+\mu$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A sequence satisfies $u_{n+2}=2u_{n+1}+3u_n+8$ for $n\ge1$, with $u_1=4$ and $u_2=4$.

   (a)

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

   [6 marks]

   (b)

   Prove by induction that $u_n=3^n-3(-1)^n-2$ for all positive integers $n$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A beekeeper models the number of bees, $B_n$ thousand, in a hive at the start of month $n$ by $B_{n+1}=0.9B_n+6$, with $B_1=20$.

   (a)

   What does the model give for $B_3$?

   [1 mark]

   • A$24$
   • B$21.6$
   • C$33.6$
   • D$27.6$

   (b)

   According to the model, what value does $B_n$ approach as $n$ becomes large?

   [1 mark]

   • A$60$
   • B$6$
   • C$54$
   • D$600$

   (c)

   Find $B_n$ in terms of $n$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A sequence satisfies $u_{n+2}-7u_{n+1}+10u_n=12$ for $n\ge1$.

   (a)

   Which expression is the complementary function, where $A$ and $B$ are constants?

   [1 mark]

   • A$A\cdot2^n+B\cdot(-5)^n$
   • B$A\cdot(-2)^n+B\cdot(-5)^n$
   • C$A\cdot2^n+B\cdot5^n$
   • D$(A+Bn)\cdot2^n$

   (b)

   The particular solution is a constant $\lambda$. What is $\lambda$?

   [1 mark]

   • A$-2$
   • B$3$
   • C$12$
   • D$6$

   (c)

   Given that $u_1=2$ and $u_2=-14$, find the general closed form for $u_n$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A sequence is defined by $u_{n+1}=4u_n-3$ for $n\ge1$, with $u_1=2$.

   (a)

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

   [3 marks]

   (b)

   Prove by induction that $u_n=4^{n-1}+1$ for all positive integers $n$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The number of seedlings, $S_n$ hundred, in a nursery in year $n$ is modelled by $S_{n+2}=S_{n+1}+2S_n$ for $n\ge1$, with $S_1=1$ and $S_2=3$.

   (a)

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

   [6 marks]

   (b)

   Let $T_n=S_{n+1}+S_n$.
   (i) Show that $T_{n+1}=2T_n$.
   (ii) Find $T_n$ in terms of $n$.
   (iii) Hence, by solving the first order recurrence relation $S_{n+1}+S_n=2^{n+1}$, find $S_n$ in terms of $n$.

   [6 marks]

   Total for question 8: 12 marks

End of questions