Q1
Answer
Write down the first 4 terms of the sequence with $n$th term given by $u_n = 2^n$.
$2,4,8,16$
Q2
Answer
The $n$th term of the sequence $$4,7,10,13, ...$$ is given by $u_n = an+b$. Find $u_n$.
$3n+1$
Q3
Answer
Suggest possible expressions for the $n$th term of the following sequences:
  1. $3,9,27,81,243...$
  2. $0,1,8,27,64...$
  1. $3^n$
  2. $(n-1)^3$
Q4
Answer
The $n$th term of a sequence is given by $$u_n = a + 3^{n-2}$$ Given that $u_3 = 11$, find $u_6$.
$89$
Q5
Answer
The $n$th term of a sequence is given by $$u_n = n(2n+k)$$ Given $u_6 = 2u_4-2$, prove that $$u_n - u_{n-1} = an+b$$ for all values of $n$, where $a$ and $b$ are constants to be found.
$4n+3$
Q6
Answer
The $n$th term of a sequence is given by $$u_n = k^n-3$$ Given that $u_1+u_2 = 0$, find two possible values for $u_5$.
$-246, 29$
Q7
Answer
Write down the first 4 terms of the following sequences:
  1. $u_n = u_{n-1}+4$, $u_1 = 3$
  2. $u_{n+1} = 2u_n+5$, $u_1 = -2$
  1. $3,7,11,15$
  2. $-2,1,7,19$
Q8
Answer
The sequence $$-4,-3,-1,3,11$$ can be defined by the recurrence relation $u_n = au_{n-1} + b$. Find $a$ and $b$.
$a=2,b=5$
Q9
Answer
For the following sequences, find expressions for $u_3$ in terms of $k$:
  1. $u_n = 4u_{n-1}-k \quad u_1 = k$
  2. $u_{n+1} = \dfrac{u_n}{k} \quad u_1 = 4$
  1. $11k$
  2. $\dfrac{4}{k^2}$
Q10
Answer
For the following sequences, find expressions for $u_3$ in terms of $k$:
  1. $u_n = 2-ku_{n-1} \quad u_1 = -1$
  2. $u_{n+1} = \sqrt[3]{61k^3+u_n^3} \quad u_1 = k\sqrt[3]{3}$
  1. $2-2k-k^2$
  2. $5k$
Q11
Answer
Given $$u_n = \frac{1}{2}(k+3u_{n-1}) \quad u_1 = 2 \quad u_3 = 7$$ find $u_4$.
$\dfrac{23}{2}$
Q12
Answer
For the following sequences, find $u_1$:
  1. $u_n = \dfrac{1}{2}\sqrt{u_{n-1}} \quad u_3 = 1$
  2. $u_{n+1} = 0.2(1-u_n) \quad u_3 = -0.2$
  1. $64$
  2. $-9$
Q13
Answer
For the following sequences, find $u_1$ and $u_4$:
  1. $u_n = 3u_{n-1}-2 \quad u_3 = 10$
  2. $u_{n+1} = \dfrac{3u_n}{4}+2 \quad u_3 = 5$
  1. $u_1 = 2$, $u_4 = 28$
  2. $u_1 = \dfrac{8}{3}$, $u_4 = \dfrac{23}{4}$
Q14
Answer
Given $$u_{n+1} = u_n + c \quad u_1 = 2 \quad u_5 = 30$$ find an expression for $u_n$ in terms of $n$.
$7n-5$
Q15
Answer
Given $$u_n = 3(u_{n-1} - k) \quad u_1 = -4 \quad u_3 = 7u_2 +3$$ find the value of $u_4$.
$u_4 = 87$
Q16
Answer
A sequence is defined by $$u_n = ku_{n-1} + 2 \quad u_1 = 1.5$$ Find the possible values of $k$ if $u_3 = 12$.
$-\dfrac{10}{3}, 2$
Q17
Answer
The first term of an arithmetic sequence is $14$. The fourth term is $32$. What is the common difference?
$6$
Q18
Answer
For an arithmetic sequence, the 20th term is $14$ and the 40th term is $-6$. Find the 10th term.
$24$
Q19
Answer
Find the number of terms of these arithmetic sequences:
  1. $90,88,..., 16,14$
  2. $x,3x,5x,...,35x$
  1. $39$
  2. $18$
Q20
Answer
A geometric sequence has first term $4$ and third term $1$. Find two possible values for the 8th term.
$\pm\dfrac{1}{32}$
Q21
Answer
The first 3 terms of an arithmetic sequence are $-8,k^2,17k$. Find two possible values for $k$.
$\dfrac{1}{2}, 8$
Q22
Answer
An arithmetic sequence has first term $x^2$ and common difference $x$, where $x>0$. The fifth term is $41$. Find $x$ in the form $p+q\sqrt{5}$.
$-2+3\sqrt{5}$
Q23
Answer
The first three terms of a geometric sequence are $9,36,144$. State whether $383,616$ is a term in the sequence.
$383616 = 2^7\times 3^4\times 37^1$, not in the sequence.
Q24
Answer
The first 3 terms of a geometric sequence are given by $$8-x \quad 2x \quad x^2$$ where $x>0$. Find the value of the 16th term.
$131072$