Sequences and recurrence relationsAQA A-Level Maths: Subtopic test

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

   (a)

   Find the value of $u_4$.

   [1 mark]

   • A$17$
   • B$33$
   • C$9$
   • D$19$

   (b)

   Which statement about the sequence is correct?

   [1 mark]

   • AThe sequence is decreasing.
   • BThe sequence is periodic with order $2$.
   • CThe sequence is increasing.
   • DThe sequence alternates above and below $3$.

   (c)

   Show that $u_n=2^n+1$ satisfies both $u_1=3$ and the relation $u_{n+1}=2u_n-1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sequence has $n$th term $a_n=\frac{n}{n+2}$ for $n\ge1$.

   (a)

   Find the value of $a_3$.

   [1 mark]

   • A$\frac34$
   • B$\frac35$
   • C$\frac53$
   • D$\frac25$

   (b)

   Which is the first term of the sequence that is greater than $0.9$?

   [1 mark]

   • A$a_{17}$
   • B$a_{18}$
   • C$a_{20}$
   • D$a_{19}$

   (c)

   Prove that the sequence is increasing.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence is defined by $x_1=2$ and $x_{n+1}=\frac{1}{1-x_n}$ for $n\ge1$.

   (a)

   Find $x_2$, $x_3$ and $x_4$, and state what this shows about the sequence.

   [3 marks]

   (b)

   Find $x_{100}$ and the sum of the first $100$ terms.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A sequence is defined by $u_1=1$ and $u_{n+1}=ku_n+4$ for $n\ge1$, where $k$ is a constant.

   (a)

   (i) Show that $u_3=(k+2)^2$.
   (ii) Given that $u_3=25$ and $k>0$, find the value of $k$.
   (iii) Hence find $u_4$.

   [6 marks]

   (b)

   Find the values of $k$ for which $u_3=u_1$, and describe the sequence for each value.
   For the value of $k$ that gives a sequence of order $2$, find the sum of the first $50$ terms.

   [6 marks]

   Total for question 4: 12 marks

End of questions