Number sequences and their rules
A fruit seller packs 5 more mangoes in each next box: 5, 10, 15, 20, … Anyone can tell what comes next, because the boxes follow a pattern. Mathematics is the search for such patterns and for the reason why they happen.
A number sequence is a list of numbers that follows a rule. The branch of mathematics that studies patterns in whole numbers is called number theory. To find the rule, look at how one number changes into the next.
In the figure: Top row: add 5 each time. Bottom row: double each time. The arrows show the rule that takes you from each number to the next one.
Rule 1: Find the rule — Check the gap between neighbouring numbers. If the gap is always the same, the rule is add (or subtract) that number. If every number is a fixed number of times the one before, the rule is multiply.
- Easy: 4, 8, 12, 16: gap 4 each time, so the next number is 16 + 4 = 20.
- Medium: 50, 44, 38, 32: the numbers go down by 6, so the next number is 32 − 6 = 26.
- Exam-type: 5, 15, 45, 135: each number is 3 times the one before, so the next is 135 × 3 = 405.
Rule 2: Famous sequences — Learn these: counting numbers (1, 2, 3, …), odd numbers (1, 3, 5, …), even numbers (2, 4, 6, …), powers of 2 (double each time), powers of 3 (3 times each time) and Virahānka numbers (each number is the sum of the two before it).
- Easy: Powers of 2: 1, 2, 4, 8, 16, so the next number is 32.
- Medium: A Virahānka-type list starts 3, 4: 3, 4, 7, 11, 18, so the next number is 11 + 18 = 29.
- Exam-type: The 15th odd number: double the position and take away 1, so 2 × 15 − 1 = 29.
A rule must work for every step of the sequence, not only for the first gap.
Guessing the rule from just two numbers: 2, 4 could be “add 2” (6, 8) or “double” (8, 16). Check at least three numbers.
First check the gap; if the gaps are not equal, check how many times; if neither works, check the sum of the two numbers before. Memory clue: “Gap – Times – Sum”: G-T-S. Ask these three questions in this order.
Spot the difference
| Point | Add rule (5, 10, 15, 20) | Multiply rule (3, 6, 12, 24) |
|---|---|---|
| Gap between numbers | always 5 | 3, 6, 12 (grows) |
| Rule | add 5 | multiply by 2 |
| Number after the last | 20 + 5 = 25 | 24 × 2 = 48 |
Practice questions
- 1What comes next? 6, 12, 18, 24, __
- 2What is the rule for 40, 35, 30, 25, …?
- 3What is the 6th even number?
- 4What comes next? 4, 12, 36, 108, __
- 5A Virahānka-type list starts 2, 5. Write the first six numbers.
- 6Find the missing number: 3, 10, __, 24, 31
- 7What comes next? 3, 5, 9, 17, 33, __ (hint: look at the gaps)
- 8Is 50 in the sequence 7, 12, 17, 22, …? Give a reason.
Show answers (try them yourself first)
- 130 (add 6)
- 2subtract 5
- 312 (6 × 2)
- 4324 (108 × 3)
- 52, 5, 7, 12, 19, 31
- 617 (add 7 each time)
- 765 (the gaps are 2, 4, 8, 16, so the next gap is 32: 33 + 32 = 65)
- 8No. The numbers go up by 5 and always end in 2 or 7; 50 ends in 0.
Picturing sequences: triangular, square, cube and hexagonal numbers
A shopkeeper piles oranges in a triangle: 1 on top, 2 in the next row, 3 in the next. A tiler lays 4 rows of 4 tiles. A child builds a cube from small blocks. Pictures like these help us see number patterns.
Triangular numbers are dots in a triangle with rows of 1, 2, 3, … dots. Square numbers are dots in a square grid (rows × columns). Cube numbers are small cubes in a big cube (length × breadth × height). Hexagonal numbers (1, 7, 19, 37, 61, …) are dots in a growing hexagon pattern.
In the figure: The purple triangle has rows of 1, 2, 3, 4 dots: 10 dots. The green square has 4 rows of 4 dots: 16 dots.
In the figure: A 3 × 3 × 3 cube is 3 layers of 3 × 3 small cubes: 27 small cubes in all.
Rule 1: Triangular numbers — To go to the next triangular number, add one more row: the new row has as many dots as its row number. So add 1, then 2, then 3, then 4, …
Picturing sequences: triangular, square, cube and hexagonal numbers
In the app you can move this figure with your finger.
Open in the app- Easy: 4th triangular number: 1 + 2 + 3 + 4 = 10.
- Medium: 7th triangular number: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
- Exam-type: 36 dots make a triangle of 8 rows (1 + 2 + … + 8 = 36). They also make a 6 × 6 square, so 36 is both triangular and square.
Rule 2: Square, cube and hexagonal numbers — Square number: a number multiplied by itself (2 times). Cube number: a number multiplied by itself 3 times. Hexagonal number: 6 times a triangular number, plus 1.
- Easy: 5th square number: 5 × 5 = 25.
- Medium: 4th cube number: 4 × 4 × 4 = 64.
- Exam-type: Is 100 a cube? 4 × 4 × 4 = 64 and 5 × 5 × 5 = 125, so 100 is not a cube (but 10 × 10 = 100 is a square). Hexagonal check: 6 × 10 + 1 = 61.
The same number can play different roles: 36 is a triangular number and a square number.
Mixing up “square of 4” and “4 × 2”: the square of 4 is 4 × 4 = 16, not 8. The cube of 3 is 27, not 9.
Triangular: keep adding 1, 2, 3, 4, … to the last number. Square: the gaps between squares are the odd numbers 3, 5, 7, 9, … Memory clue: Square = 2 times (flat picture), Cube = 3 times (solid picture): the number of times matches the dimensions, 2D and 3D.
Spot the difference
| Point | Triangular | Square | Cube |
|---|---|---|---|
| How it is made | add rows 1, 2, 3, … | number × number | number × number × number |
| First four | 1, 3, 6, 10 | 1, 4, 9, 16 | 1, 8, 27, 64 |
| Picture | triangle of dots | square of dots | cube of blocks |
Practice questions
- 1What is the 6th triangular number?
- 2What is the 9th square number?
- 3What is the 3rd cube number?
- 4How many new dots are added to go from the 7th to the 8th triangular number?
- 5Which of 28, 30, 32 is a triangular number?
- 6What is the 10th square number minus the 9th square number?
- 7Which is bigger, the 10th triangular number or the 7th square number, and by how much?
- 8Find the two hexagonal numbers that come after 61 (use 6 × triangular number + 1).
Show answers (try them yourself first)
- 121 (1 + 2 + 3 + 4 + 5 + 6)
- 281 (9 × 9)
- 327 (3 × 3 × 3)
- 48 (the new row is the 8th row)
- 528 (1 + 2 + … + 7 = 28)
- 6100 − 81 = 19 (the next odd number)
- 710th triangular = 55, 7th square = 49. The triangular number is bigger by 55 − 49 = 6.
- 891 (6 × 15 + 1) and 127 (6 × 21 + 1)
Relations among number sequences
A tiler starts with 1 tile, then lays an L-shaped border of 3 tiles, then a border of 5, then 7. Each time the floor is still a perfect square. Different sequences are often linked in surprising ways, and a picture can show why.
Two sequences are related when one can be made from the other by a fixed rule. Finding why the relation works (not only that it works) is the real fun of mathematics, and a picture often gives the reason.
In the figure: Each colour is one L-shaped layer. The layers have 1, 3, 5, 7, 9, 11 dots and together make a 6 × 6 square.
In the figure: Each slanting coloured line holds 1, 2, 3, 4, 5, 4, 3, 2, 1 dots: together a 5 × 5 square.
Rule 1: Odd numbers add up to a square — Adding the first few odd numbers always gives a square number: the answer is (how many odd numbers you added) × (the same count).
Relations among number sequences
In the app you can move this figure with your finger.
Open in the app- Easy: 1 + 3 + 5 = 9 = 3 × 3 (3 odd numbers).
- Medium: The first 9 odd numbers add up to 9 × 9 = 81.
- Exam-type: The first 40 odd numbers give 40 × 40 = 1,600. If the sum is 225 = 15 × 15, then 15 odd numbers were added.
Rule 2: Adding up and down, and other relations — Adding 1, 2, 3, … up to a top number and back down again gives (top number) × (top number). Other links: two neighbouring triangular numbers add to a square; 6 × triangular + 1 is hexagonal; 1 + 2 + 4 + … plus 1 is the next power of 2.
Relations among number sequences
In the app you can move this figure with your finger.
Open in the app- Easy: 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4 × 4.
- Medium: Neighbouring triangular numbers: 15 + 21 = 36 = 6 × 6.
- Exam-type: Up to 25 and back down: 25 × 25 = 625. Also 1 + 2 + 4 + 8 + 16 = 31 and 31 + 1 = 32, the next power of 2.
A relation is only complete when we know why it happens: the L-shaped picture explains why odd numbers build squares for every size.
Using the last odd number instead of the count: the first 5 odd numbers end at 9, but the sum is 5 × 5 = 25, not 9 × 9.
Odd numbers: count them, then multiply the count by itself. Up and down: take the top number and multiply it by itself. Memory clue: “Count × Count” for odd numbers, “Top × Top” for up-and-down: C-C and T-T.
Spot the difference
| Point | Odd numbers added | Added up and down |
|---|---|---|
| Example | 1 + 3 + 5 + 7 | 1 + 2 + 3 + 2 + 1 |
| What to multiply | count 4 × 4 | top 3 × 3 |
| Answer | 16 | 9 |
Practice questions
- 1Find 1 + 3 + 5 + 7 + 9.
- 2Find 1 + 2 + 3 + 4 + 3 + 2 + 1.
- 3Add the triangular numbers 6 and 10.
- 4What is the sum of the first 20 odd numbers?
- 5Find 1 + 2 + 3 + … + 10 + … + 3 + 2 + 1.
- 6Multiply the triangular number 15 by 6 and add 1. Which kind of number is it?
- 7The sum of the first few odd numbers is 169. How many odd numbers were added?
- 8Find 1 + 2 + 4 + 8 + 16 + 32 + 64 and then add 1. Which power of 2 do you get?
Show answers (try them yourself first)
- 125 (5 × 5)
- 216 (4 × 4)
- 316 (a square number)
- 4400 (20 × 20)
- 5100 (10 × 10)
- 66 × 15 + 1 = 91, a hexagonal number
- 713 (169 = 13 × 13)
- 81 + 2 + 4 + 8 + 16 + 32 + 64 = 127; 127 + 1 = 128, which is the next power of 2 after 64
Shape sequences and regular polygons
A honeycomb has 6-sided cells, a stop sign has 8 sides and a carrom board has 4. Shapes also come in sequences: triangle, quadrilateral, pentagon, … each with one more side than the one before.
The branch of mathematics that studies patterns in shapes is called geometry. A polygon is a closed shape with straight sides. A polygon is regular when all its sides are equal and all its corners (angles) are equal. In every closed shape the number of sides equals the number of corners.
In the figure: Regular polygons with 3, 4, 5, 6, 7 and 8 sides: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon.
Rule 1: Name from the number of sides — The name tells the number of sides: tri 3, quadri 4, penta 5, hexa 6, hepta 7, octa 8, nona 9, deca 10. In the list that starts with the triangle, the number of sides goes up by 1 each time.
Shape sequences and regular polygons
In the app you can move this figure with your finger.
Open in the app- Easy: A shape with 6 sides is a hexagon.
- Medium: The 7th shape in the list triangle, quadrilateral, pentagon, … has 9 sides: a nonagon.
- Exam-type: The 12th shape has 3 + 11 = 14 sides.
Rule 2: Perimeter of a regular polygon — All sides are equal, so the perimeter = (number of sides) × (length of one side).
- Easy: Regular pentagon, side 4 cm: 5 × 4 = 20 cm.
- Medium: Regular octagon, side 7 cm: 8 × 7 = 56 cm.
- Exam-type: Regular nonagon with perimeter 72 cm: each side = 72 ÷ 9 = 8 cm.
Corners = sides in every closed figure, so a polygon with 10 sides also has 10 corners.
Calling every shape with equal sides “regular”. A rhombus has equal sides but unequal angles; a 6 cm × 3 cm rectangle has equal angles but unequal sides. Neither is regular.
Regular quadrilateral = square. Regular triangle = equilateral triangle. Memory clue: Octopus has 8 legs, so octagon = 8. A decade is 10 years, so decagon = 10. Tri, Quad, Pent, Hex, Hept, Oct, Non, Dec = 3 to 10.
Spot the difference
| Point | Square | Rectangle (6 × 3) | Rhombus |
|---|---|---|---|
| All sides equal | yes | no | yes |
| All corners equal | yes | yes | no |
| Regular? | yes | no | no |
Practice questions
- 1How many sides does a nonagon have?
- 2What is a polygon with 7 sides called?
- 3How many corners does a hexagon have?
- 4A regular hexagon has each side 9 cm. Find its perimeter.
- 5Which is the 5th shape in the list triangle, quadrilateral, pentagon, …?
- 6A regular decagon has each side 6 cm. Find its perimeter.
- 7A regular polygon has perimeter 56 cm and each side 8 cm. Which polygon is it?
- 8A rectangle has all angles 90°. Is it a regular polygon? Give a reason.
Show answers (try them yourself first)
- 19
- 2heptagon
- 36
- 46 × 9 = 54 cm
- 5heptagon (7 sides)
- 610 × 6 = 60 cm
- 756 ÷ 8 = 7 sides: a heptagon
- 8Not unless it is a square. Its sides are not all equal (for example 6 cm and 3 cm), so it is not regular.
Shape sequences linked to number sequences
At a party, 5 friends meet and every friend shakes hands once with each of the others. Draw 5 points and join every pair: the lines are the handshakes. Counting such lines links a shape to a number sequence.
A complete graph joins every two of its points by a line. Stacked squares and stacked triangles are shapes built row by row. The Koch snowflake starts as a triangle and replaces every line segment by a “speed bump” of 4 smaller segments, again and again.
In the figure: Complete graphs with 2, 3, 4, 5 and 6 points have 1, 3, 6, 10 and 15 lines: the triangular numbers.
In the figure: Stacked squares have 1, 4, 9, 16 little squares: the square numbers.
Rule 1: Counting lines in a complete graph — The first point joins to all the others; the next point has one fewer new line; and so on. So the lines = (points − 1) + (points − 2) + … + 1, which gives the triangular numbers.
Shape sequences linked to number sequences
In the app you can move this figure with your finger.
Open in the app- Easy: 4 points: 3 + 2 + 1 = 6 lines.
- Medium: 6 points: 5 + 4 + 3 + 2 + 1 = 15 lines.
- Exam-type: 10 friends shaking hands: 9 + 8 + … + 1 = 45 handshakes.
Rule 2: Stacked shapes give square numbers — Stacked squares: rows × columns gives (number) × (same number). Stacked triangles: the rows have 1, 3, 5, 7, … triangles, and odd numbers add up to a square.
Shape sequences linked to number sequences
In the app you can move this figure with your finger.
Open in the app- Easy: 5th stacked square: 5 × 5 = 25 little squares.
- Medium: Stacked triangles with 7 rows: 7 × 7 = 49 little triangles.
- Exam-type: The 9th row has 2 × 9 − 1 = 17 triangles, and 9 rows give 9 × 9 = 81 in all.
Rule 3: Koch snowflake: multiply by 4 — Every line segment becomes 4 segments, so the count of segments is multiplied by 4 each time: 3, 12, 48, 192, … (3 times the powers of 4).
Shape sequences linked to number sequences
In the app you can move this figure with your finger.
Open in the app- Easy: 2nd shape: 3 × 4 = 12 segments.
- Medium: 4th shape: 3 × 4 × 4 × 4 = 192 segments.
- Exam-type: 6th shape: 3 × 4 × 4 × 4 × 4 × 4 = 3,072 segments.
A shape sequence and a number sequence can be the same pattern in two forms: dots, lines or squares counted give the numbers.
Counting the lines of a complete graph as points × (points − 1): for 5 points that gives 20, but every line is counted twice. The real answer is 20 ÷ 2 = 10.
Handshakes: add (people − 1) down to 1. Stacked shapes: (rows) × (rows). Koch: keep multiplying by 4. Memory clue: “Handshake countdown”: 4, 3, 2, 1 — the last person has no new handshake. Koch = “×4 each time”.
Spot the difference
| Shape sequence | Number sequence it gives | Example |
|---|---|---|
| Regular polygons (sides) | counting numbers from 3 | 3, 4, 5, 6 |
| Complete graphs (lines) | triangular numbers | 1, 3, 6, 10 |
| Stacked squares / triangles | square numbers | 1, 4, 9, 16 |
| Koch snowflake (segments) | 3 × powers of 4 | 3, 12, 48, 192 |
Practice questions
- 1How many lines does a complete graph with 5 points have?
- 2How many little squares are in the 6th stacked square?
- 3How many line segments does the first Koch snowflake shape have?
- 47 friends meet and each shakes hands once with every other friend. How many handshakes?
- 5How many little triangles are in stacked triangles with 8 rows?
- 6How many line segments does the 3rd Koch snowflake shape have?
- 7In a group, the handshakes (each pair once) total 66. How many people are in the group?
- 8How many line segments does the 5th Koch snowflake shape have?
Show answers (try them yourself first)
- 110 (4 + 3 + 2 + 1)
- 236 (6 × 6)
- 33 (it is a triangle)
- 421 (6 + 5 + 4 + 3 + 2 + 1)
- 564 (8 × 8)
- 648 (12 × 4)
- 712 people (11 + 10 + … + 1 = 66)
- 8768 (3 × 4 × 4 × 4 × 4)