Sequences, recurrence relations and sigma notationEdexcel A-Level Maths: Subtopic test

1. 1
   A sequence is defined by $u_1=5$ and $u_{n+1}=2u_n-3$ for $n\ge1$.

   (a)

   Find the value of $u_4$.

   [1 mark]

   • A$11$
   • B$19$
   • C$40$
   • D$35$

   (b)

   Which statement about the sequence is correct?

   [1 mark]

   • AIt is decreasing.
   • BIt is periodic.
   • CIt converges to $3$.
   • DIt is increasing.

   (c)

   Calculate $\sum_{r=1}^{4}u_r$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sequence is defined by $u_1=2$ and $u_{n+1}=\frac{8}{u_n}$ for $n\ge1$.

   (a)

   Find the value of $u_{20}$.

   [1 mark]

   • A$4$
   • B$2$
   • C$8$
   • D$\frac{2}{5}$

   (b)

   Find the value of $\sum_{r=1}^{50}u_r$.

   [1 mark]

   • A$300$
   • B$100$
   • C$150$
   • D$200$

   (c)

   Explain why the sequence is periodic and state its order.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence has $n$th term $u_n=n^2-8n+20$ for $n\ge1$.

   (a)

   Show that $u_{n+1}-u_n=2n-7$, and hence find the values of $n$ for which $u_{n+1}<u_n$.

   [3 marks]

   (b)

   Find the value of $\sum_{r=1}^{7}\left(u_r-4\right)$, using the result $\sum_{r=1}^{n}1=n$ where appropriate.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A sequence is defined by $u_1=2$ and $u_{n+1}=\frac{1}{1-u_n}$ for $n\ge1$.

   (a)

   (i) Find $u_2$, $u_3$ and $u_4$.
   (ii) Explain why the sequence is periodic and state its order.
   (iii) Find $u_{100}$.

   [6 marks]

   (b)

   (i) Find $\sum_{r=1}^{100}u_r$.
   (ii) Find the least value of $N$ for which $\sum_{r=1}^{N}u_r>200$.

   [6 marks]

   Total for question 4: 12 marks

End of questions