(a) A sequence of shapes is made with squares and triangles.
(i) On the grid, draw Shape 3.
(ii) Find the number of triangles in Shape 5.
(b) These are the first four terms of a sequence.
32 25 18 11
(i) Find the next two terms.
(ii) Write down the term to term rule for this sequence.
(c) (i) 5, 8, 11, 14, ……
Find the nth term of this sequence.
(ii) 1, 8, 27, 64, ……
Find the nth term of this sequence.
▶️ Answer/Explanation
(a)(i) 
Shape 3 should have 3 squares and 6 triangles.
(a)(ii) 12 triangles (pattern is 3×n triangles for Shape n).
(b)(i) Next terms are 4, -3 (subtracting 7 each time).
(b)(ii) Term-to-term rule: Subtract 7.
(c)(i) nth term: \(3n + 2\) (starts at 5 and increases by 3 each time).
(c)(ii) nth term: \(n^3\) (these are cube numbers).
These are the first four diagrams in a sequence. The diagrams are made using dots and lines.
(a) Complete the table.
| Diagram | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Number of small squares | 2 | 4 | 6 | |
| Number of dots | 6 | 9 | 12 | |
| Number of lines | 7 | 12 | 17 |
(b) Complete this statement.
A diagram in this sequence cannot have 51 small squares because ……
(c) An expression for the number of dots in Diagram n is 3n + 3.
Which diagram has 249 dots?
(d)(i) Find an expression, in terms of n, for the number of lines in Diagram n.
(ii) Find the number of lines in Diagram 41.
▶️ Answer/Explanation
(a)
Number of small squares: 8 (pattern increases by 2 each time)
Number of dots: 15 (pattern increases by 3 each time)
Number of lines: 22 (pattern increases by 5 each time)
(b)
51 is an odd number, and the sequence only has even numbers of squares.
(c)
Diagram 82 (solve 3n + 3 = 249 → 3n = 246 → n = 82)
(d)(i)
5n + 2 (the pattern shows each term increases by 5 from the previous one)
(d)(ii)
207 (substitute n=41 into 5n+2: 5×41 + 2 = 205 + 2 = 207)