Sequences and recurrence relationsEdexcel International A Level Maths: Subtopic test

1. 1
   A sequence is given by $u_n=n^2-6n+10$ for $n\ge1$.

   (a)

   Find the value of $u_5$.

   [1 mark]

   • A$2$
   • B$29$
   • C$65$
   • D$5$

   (b)

   Which statement about the sequence is correct?

   [1 mark]

   • AIt is neither increasing, decreasing nor periodic
   • BIt is an increasing sequence
   • CIt is a decreasing sequence
   • DIt is periodic with order 2

   (c)

   Find an expression for $u_{n+1}-u_n$ in terms of $n$, and hence state the values of $n$ for which $u_{n+1}>u_n$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Find the value of $u_3$.

   [1 mark]

   • A$-1$
   • B$\frac12$
   • C$-\frac12$
   • D$2$

   (b)

   The sequence is periodic. What is its order, the number of terms in one repeating cycle?

   [1 mark]

   • A$1$
   • B$2$
   • C$3$
   • D$4$

   (c)

   Find the value of $u_{100}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence has $n$th term $u_n=a+b(-1)^n$, where $a$ and $b$ are constants, $u_1=3$ and $u_2=9$.

   (a)

   Find the values of $a$ and $b$.

   [3 marks]

   (b)

   Find $u_3$ and $u_4$, state the order of the sequence, and find $u_{101}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A sequence $u_1,u_2,u_3,\dots$ is defined by $u_{n+1}=pu_n+q$, where $p$ and $q$ are constants. Given that $u_1=2$, $u_2=7$ and $u_3=22$.

   (a)

   (i) Find the values of $p$ and $q$.
   (ii) Find $u_5$.

   [6 marks]

   (b)

   Given that $p=3$ and $q=1$, (i) prove that the sequence is increasing, (ii) find the smallest value of $n$ for which $u_n>10\,000$.

   [6 marks]

   Total for question 4: 12 marks

End of questions