1. 1
   A conveyor-belt sorting machine at a factory stamps parts with identification numbers following a fixed sequence. The first five numbers stamped are 5, 9, 13, 17, 21.

   (a)

   What is the term-to-term rule for this sequence?

   [1 mark]

   • AAdd 4 to the previous term
   • BAdd 3 to the previous term
   • CMultiply the previous term by 2
   • DAdd 5 to the previous term

   (b)

   Which expression gives the position-to-term (nth term) rule for this sequence?

   [1 mark]

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

   (c)

   The machine's second production run instead stamps parts with the triangular numbers: 1, 3, 6, 10, 15, ... What is the 6th triangular number?

   [1 mark]

   • A18
   • B20
   • C25
   • D21

   Total for question 1: 3 marks

2. 2
   A landscape gardener builds a patio using square paving slabs arranged in an expanding linear pattern. Pattern 1 uses 6 slabs, Pattern 2 uses 10 slabs, Pattern 3 uses 14 slabs, with each pattern using 4 more slabs than the last.

   (a)

   Find an expression, in terms of $n$, for the number of slabs used in Pattern $n$.

   [2 marks]

   (b)

   The gardener has exactly 50 slabs available. Using your expression from part (a), find the largest pattern number she can build without exceeding her supply of slabs.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A textile designer creates a diamond-tile pattern where pattern number $n$ uses $T(n)$ tiles. Pattern 1 uses 3 tiles, Pattern 2 uses 8 tiles, Pattern 3 uses 15 tiles, and Pattern 4 uses 24 tiles.

   (a)

   Find an expression for $T(n)$, the number of tiles in pattern $n$.

   [3 marks]

   (b)

   Determine, showing full algebraic working, whether 200 is a term in this sequence.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A savings scheme pays compound interest so that an initial deposit of £200 grows each year. The balance after year 1 is £220, after year 2 is £242, and after year 3 is £266.20.

   (a)

   Show that these balances form a geometric progression with common ratio 1.1, and hence write down an expression for the balance, $B(n)$, after $n$ complete years.

   [4 marks]

   (b)

   Using your expression for $B(n)$, determine after how many complete years the balance will first exceed £350.

   [4 marks]

   (c)

   A rival scheme instead pays out a linear (arithmetic) sequence of yearly balances: £200 in year 1, £215 in year 2, £230 in year 3, increasing by £15 each year.
   
   Find an expression for the balance in year $n$, then determine algebraically, showing full working, whether this scheme's balance will ever be exactly £400.

   [5 marks]

   Total for question 4: 13 marks

End of questions