Some sequences are worth recognizing on sight rather than working out
Square numbers
1, 4, 9, 16, 25, … are the square numbers. The square number in position n is n × n, written , and it is the number of tiles in a square with n tiles along each side.
Going from one square to the next adds a border of tiles along two sides. From 9 to 16 the border is 3 tiles, another 3 tiles and 1 in the corner, which is 7. So the gaps between the square numbers are the odd numbers 3, 5, 7, 9, …
The 4 by 4 square holds 16 tiles. The 3 by 3 square inside it holds 9, and the border around it adds 7.
√16 =
Grow the square one row and one column at a time. The number of tiles goes 1, 4, 9, 16 and 25: these are the square numbers.
Triangle numbers
1, 3, 6, 10, 15, … are the triangle numbers. The triangle number in position n is the number of dots in a staircase with 1 dot in its first column, 2 in the next, and so on up to n.
Each step adds one more dot than the step before: add 2, then 3, then 4. So the triangle number in position n is 1 + 2 + 3 + … + n, and the 5th is 10 + 5 = 15.
Each term is the one before with a new column one dot taller, so the gaps are 2, 3 and 4.
Two staircases make a rectangle
Take two copies of the same staircase and turn one upside down. They fit together into a rectangle n dots high and n + 1 dots wide, which holds n(n + 1) dots. One staircase is half of the rectangle, so the triangle number in position n is half of n(n + 1).
For n = 4 that is 4 × 5 = 20, and half of 20 is 10. For n = 20 it is 20 × 21 = 420, and half of 420 is 210, found without adding twenty numbers.
Set n = 6 and slide the second staircase in
Slide the second staircase into the first. Together they make a rectangle n by n + 1, so one staircase is half of n(n + 1).
The Fibonacci sequence
In the Fibonacci sequence each term is the sum of the two terms before it. It starts 1, 1, and then 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8 and 5 + 8 = 13.
The rule needs two terms to make the next one, so the sequence has to be given its first two terms before the rule can start.
Each jump along the line is the term two places back: from 8 to 13 the jump is 5.
The usual mistakes
Squaring is not doubling: the 6th square number is 6 × 6 = 36, not 6 × 2 = 12.
A triangle number is not a square number: the 6th triangle number is half of 6 × 7, which is 21, while the 6th square number is 36.
Fibonacci adds the two terms before, not the last term twice: after 5, 8 comes 5 + 8 = 13, not 8 + 8 = 16.
Patterns and page numbers · Square Arrays of Tiles: Figure 25, and the Figure with 1089 Tiles
QuestionA pattern of square tile arrays is shown: Figure 1 has a total of 4 tiles (2 × 2). Figure 2 has 9 tiles (3 × 3). Figure 3 has 16 tiles (4 × 4). (a) How many tiles are in Figure 25? (b) Which figure number consists of exactly 1089 tiles?
1.Figure 1 is a 2 × 2 square, Figure 2 is 3 × 3, and Figure 3 is 4 × 4.
Figure 1 is 2 × 2, Figure 2 is 3 × 3, Figure 3 is 4 × 4. 2.Each side is one tile longer than the figure number: Figure n is a square of side n + 1.
The side is always one more than the figure number. 3.(a) Figure 25 has a side of 25 + 1 = 26 tiles, so it has 26 × 26 = 676 tiles.
(a) Figure 25 has side 26: 26 × 26 = 676 tiles. 4.(b) A square of 1089 tiles has a side that multiplied by itself makes 1089. The number ends in 9, so the side ends in 3 or 7, and 33 × 33 = 1089: the side is 33.
(b) 1089 tiles is a 33 × 33 square. 5.The side is one more than the figure number, so this is Figure 33 − 1 = 32.
Side 33 means Figure 33 − 1 = 32.
Answer: (a) 676 tiles; (b) Figure 32
Watch out for
- Squaring the figure number: 25 × 25 = 625. Figure 25's side is 26 tiles, not 25.
- Answering Figure 33. 33 is the side of the square; the figure number is one less, 32.
Patterns and page numbers · A Staircase of Dots: Figure 20, and the Figure with 120 Dots
QuestionDots are arranged in a triangular staircase formation: Figure 1 has 1 dot. Figure 2 has 3 dots (1 + 2). Figure 3 has 6 dots (1 + 2 + 3). Figure 4 has 10 dots (1 + 2 + 3 + 4). (a) How many dots are there in Figure 20? (b) If a figure has 120 dots, what is its figure number?
1.Figure 1 and its copy make a rectangle 1 by 2: 2 dots, so Figure 1 has 2 ÷ 2 = 1 dot.
Figure 1 with its mirror copy: a 1 × 2 rectangle, 2 dots, so the figure is 1. 2.Figure 2 and its copy make a rectangle 2 by 3: 6 dots, so Figure 2 has 6 ÷ 2 = 3 dots.
Figure 2 doubled is 2 × 3 = 6, so the figure is 3. 3.Figure 3 and its copy make a rectangle 3 by 4: 12 dots, so Figure 3 has 12 ÷ 2 = 6 dots. Figure n and its copy always make a rectangle n by n + 1.
Figure 3 doubled is 3 × 4 = 12, so the figure is 6. A figure and its copy always make n × (n + 1). 4.(a) Figure 20 and its copy make 20 × 21 = 420 dots, so Figure 20 has 420 ÷ 2 = 210 dots.
(a) Figure 20 doubled: 20 × 21 = 420, so 210 dots. 5.(b) A figure of 120 dots and its copy make a rectangle of 120 × 2 = 240 dots.
(b) 120 dots doubled is 240, a rectangle of two consecutive sides. 6.The rectangle's sides are two consecutive numbers that multiply to 240: 15 × 16 = 240. The figure number is the shorter side, 15.
15 × 16 = 240: the smaller side is the figure, 15.
Answer: (a) 210 dots; (b) Figure 15
Watch out for
- Adding the same step every time. The counts go up by 2, then 3, then 4, so a rule such as 1 + 19 × 2 for Figure 20 does not hold.
- In (b), looking for two consecutive numbers that multiply to 120. The figure is half of the rectangle, so the rectangle's sides multiply to 2 × 120 = 240.
Your turn
1.1, 1, 2, 3, 5, 8, 13, … what comes next?