1. 1
   A tiler lays tile patterns whose tile counts form a sequence.

   (a)

   A tiler lays patterns where the number of tiles used follows the sequence $5, 9, 13, 17, \ldots$. What is the term-to-term rule?

   [1 mark]

   • AMultiply the previous term by 2
   • BAdd 4 to the previous term
   • CAdd 5 to the previous term
   • DSubtract 4 from the previous term

   (b)

   Using the tile sequence $5, 9, 13, 17, \ldots$, what is the next term?

   [1 mark]

   • A22
   • B20
   • C21
   • D25

   (c)

   Find the $n$th term formula for the tile sequence $5, 9, 13, 17, \ldots$.

   [1 mark]

   • A$4n - 1$
   • B$4n + 5$
   • C$5n + 4$
   • D$4n + 1$

   Total for question 1: 3 marks

2. 2
   A stadium arranges seats in blocks whose seat counts form a quadratic sequence.

   (a)

   A stadium arranges seats in blocks, where the number of seats in successive blocks forms the sequence $2, 5, 10, 17, 26, \ldots$. Find the differences between consecutive terms, and hence the second differences.

   [2 marks]

   (b)

   Using the constant second difference found in part (a), find the $n$th term of the sequence $2, 5, 10, 17, 26, \ldots$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sculptor stacks cubes in a pattern described by a cubic sequence using subscript notation.

   (a)

   A sculptor stacks cubes to form a pattern where the total number of small cubes used follows $T_n = n^3$. Using subscript notation, find $T_1$, $T_2$, $T_3$ and $T_4$.

   [3 marks]

   (b)

   The sculptor modifies the design so that the total number of cubes follows $S_n = 2n^3 - 1$. Find $S_5$, and determine the value of $n$ for which $S_n = 127$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A biologist studies an exponentially growing bacteria population and compares it with a linear control sample.

   (a)

   A biologist records the population of bacteria doubling each hour, starting with 3 bacteria, following the sequence $3, 6, 12, 24, \ldots$. Recognise the type of sequence and find a formula for the $n$th term, $T_n$.

   [4 marks]

   (b)

   Use the formula $T_n = 3 \times 2^{n-1}$ to find the number of bacteria after 7 hours (i.e. $T_8$, since $T_1$ represents the count at hour 0).

   [4 marks]

   (c)

   A control sample of bacteria instead grows linearly, following $U_n = 3 + 15(n-1)$. Find the value of $n$ for which the exponential population $T_n = 3 \times 2^{n-1}$ first exceeds the control population $U_n$, by testing successive values of $n$ and showing your working clearly.

   [5 marks]

   Total for question 4: 13 marks

End of questions