Patterns with adding and subtracting
Every Sunday Meena saves ₹ 5 more than the week before: ₹ 7, ₹ 12, ₹ 17, ₹ 22. She follows a rule, so you can say what comes next without asking her. That is a number pattern.
A pattern (नमुना) is a list of numbers or shapes that follows a rule. To find the rule, look at the difference between neighbours. If the difference is the same each time, the rule is "add" or "subtract" that number.
In the figure : The rule add 3: 3, 6, 9, 12, 15.
Rule 1: Find the rule — Subtract a number from the next one. If the answer is the same everywhere, that is the rule.
- Easy : 7, 12, 17, 22 → add 5, so next is 27.
- Medium : 90, 83, 76, 69 → subtract 7, so next is 62.
- Exam-type : 5, 9, __, 17 → add 4, so the missing number is 13.
Rule 2: Far numbers — If the first number equals the rule number, nth number = first × n. Otherwise nth = first + (n − 1) × rule.
- Easy : 6, 12, 18, ... the 10th number is 6 × 10 = 60.
- Medium : 5, 8, 11, 14, ... the 10th number is 5 + 9 × 3 = 32.
- Exam-type : Check: 32 − 5 = 27 and 27 ÷ 9 = 3, the rule number.
A pattern may go up or down. The rule can be add or subtract.
Looking at only the first two numbers. Test the rule on all neighbours.
Memory clue: Next minus this = the rule.
Spot the difference
| Pattern | Rule |
|---|---|
| 7, 12, 17 | add 5 |
| 90, 83, 76 | subtract 7 |
| 6, 12, 18 | add 6 |
Practice questions
- 1Find the next number: 7, 12, 17, 22, __.
- 2Find the next number: 90, 83, 76, 69, __.
- 3Find the next number: 4, 8, 12, 16, __.
- 4Find the missing number: 5, 9, __, 17.
- 5Find the 10th number of 6, 12, 18, ...
- 6Find the 10th number of 5, 8, 11, 14, ...
- 7A tank has 100 litres and loses 8 litres every hour. Write the first five numbers.
- 8Raju says the rule of 2, 5, 9, 14 is add 3. Is he right?
Show answers (try them yourself first)
- 127
- 262
- 320
- 413
- 560
- 65 + 9 × 3 = 32
- 7100, 92, 84, 76, 68
- 8No. The differences are 3, 4, 5, so the gap grows by 1 each time.
Patterns with multiplying and dividing
A rumour spreads: 1 person tells 3, each of them tells 3 more. The numbers 1, 3, 9, 27 grow by multiplying. Some things shrink by dividing, like a pizza cut in half again and again.
In some patterns each number is multiplied by the same number to get the next. In others each is divided. Find the rule by dividing a number by the one before it.
In the figure : Doubling: 2, 4, 8, 16, 32.
Rule 1: Multiplying patterns — Next number ÷ this number = the multiplier.
- Easy : 5, 10, 20, 40, 80 doubles each time.
- Medium : 2, 6, 18, 54 multiplies by 3.
- Exam-type : Pocket money ₹ 1 doubled each day: day 10 is ₹ 512 (1 doubled 9 times).
Rule 2: Dividing and mixed patterns — Dividing: each number is divided by the same number. Mixed: two steps each time.
- Easy : 640, 320, 160, 80, 40, 20 halves each time (÷ 2).
- Medium : 243, 81, 27, 9 is divided by 3; next is 3.
- Exam-type : "Multiply by 3, subtract 1" starting at 2: 2, 5, 14, 41.
Multiplying patterns grow very fast. A plant doubling 5, 10, 20, 40 cm in four weeks shows it.
Using add when the pattern multiplies. 2, 6, 18 is not "add 4".
Memory clue: Same difference → add/subtract; same ratio → multiply/divide.
Spot the difference
| Pattern | Rule |
|---|---|
| 5, 10, 20, 40 | × 2 |
| 243, 81, 27, 9 | ÷ 3 |
| 2, 5, 14, 41 | × 3, − 1 |
Practice questions
- 1Find the next number: 5, 10, 20, 40, __.
- 2Find the next number: 2, 6, 18, 54, __.
- 3Find the next number: 243, 81, 27, 9, __.
- 4Find the rule: 640, 320, 160, 80.
- 5A plant is 5 cm tall and doubles its height every week. What is its height after 4 weeks?
- 6"Multiply by 3, subtract 1" starts at 2. Write four numbers.
- 7₹ 1 is doubled each day. How much on day 10?
- 8Sita says 2, 6, 18 follows "add 4". Correct her.
Show answers (try them yourself first)
- 180
- 2162
- 33
- 4Divide by 2
- 55, 10, 20, 40, 80 → 80 cm after 4 weeks
- 62, 5, 14, 41
- 7₹ 512
- 82 + 4 = 6 but 6 + 4 = 10, not 18. The rule is × 3.
Growing shape patterns
Making a row of rangoli sticks or matchstick houses: each new house needs a few more sticks. If you know how many each new one adds, you can find any figure without drawing it.
To find the rule of a growing shape pattern: count how much each figure grows (the step), then fix the first figure. Sticks = step × figure number + (first − step).
In the figure : Matchstick triangles: 3, 5, 7 sticks, so each triangle adds 2.
Rule 1: Triangles in a row — Sticks = 2 × n + 1.
- Easy : 15 triangles need 2 × 15 + 1 = 31 sticks.
- Medium : 12 triangles need 25 sticks.
- Exam-type : 51 sticks make how many triangles? 2n + 1 = 51, so n = 25.
Rule 2: Other shapes in a row — Pentagons: 5, 9, 13 → 4n + 1. Hexagons: 6, 11, 16 → 5n + 1.
- Easy : 10 pentagons need 4 × 10 + 1 = 41 sticks.
- Medium : 10 hexagons need 5 × 10 + 1 = 51 sticks.
- Exam-type : 41 sticks make (41 − 1) ÷ 4 = 10 pentagons.
Growing shape patterns
In the app you can move this figure with your finger.
Open in the appDot patterns: 1, 3, 5, 7 (L-shapes) follow 2n − 1; 1, 4, 9, 16 follow n × n.
Forgetting the "+1" because the first shape uses all its sticks. Test the rule on figure 1.
Memory clue: Find the step, then test on figure 1.
Spot the difference
| Shapes in a row | Sticks for n |
|---|---|
| Triangles | 2n + 1 |
| Pentagons | 4n + 1 |
| Hexagons | 5n + 1 |
Practice questions
- 1How many sticks make 15 triangles in a row?
- 2How many sticks make 10 pentagons in a row?
- 3How many sticks make 10 hexagons in a row?
- 4How many sticks make 25 pentagons?
- 551 sticks make how many triangles?
- 641 sticks make how many pentagons?
- 7Squares in a row use 4, 7, 10, 13 sticks. Write the rule and find sticks for 20 squares.
- 8Anil says 10 triangles need 2 × 10 = 20 sticks. Correct him.
Show answers (try them yourself first)
- 131
- 241
- 351
- 4101
- 525
- 610
- 73n + 1; 3 × 20 + 1 = 61
- 8He forgot the +1: 2 × 10 + 1 = 21 sticks.
Repeating patterns
The traffic light goes red, yellow, green, red, yellow, green again and again. If you know the group, you can tell which light comes after 100 changes without waiting.
A repeating pattern has a group of items that comes again and again. To find the item at a far position, divide the position by the group size. The remainder tells the place in the group. If the remainder is 0, it is the last item of the group.
In the figure : Four colours repeating: position 15 is 12 + 3, the third colour.
Rule 1: Position ÷ group size — The remainder gives the place in the group.
- Easy : R G B Y repeated: 22 ÷ 4 = 5 remainder 2, so the 22nd is G.
- Medium : The 40th: 40 ÷ 4 = 10 remainder 0, so the last item, Y.
- Exam-type : Beads ● ● ○ repeated: the 50th bead is 50 ÷ 3 = 16 remainder 2, so ●.
Rule 2: Calendar patterns — Weekdays repeat every 7 days. Months repeat every 12.
- Easy : Friday + 20 days: 20 ÷ 7 = 2 remainder 6, so Friday + 6 = Thursday.
- Medium : Sunday + 45 days: 45 ÷ 7 = 6 remainder 3, so Wednesday.
- Exam-type : January + 25 months: 25 ÷ 12 = 2 remainder 1, so February.
The remainder is the place in the group. A remainder of 0 means the last item.
Using the quotient instead of the remainder. 22 ÷ 4 = 5 is not the place.
Memory clue: Remainder tells the place, zero is the last.
Spot the difference
| Position | Item in R G B Y |
|---|---|
| 22 | G (remainder 2) |
| 25 | R (remainder 1) |
| 40 | Y (remainder 0) |
Practice questions
- 1In R G B Y repeated, what is the 22nd?
- 2In R G B Y repeated, what is the 25th?
- 3In R G B Y repeated, what is the 40th?
- 4Beads red, blue, green repeat. What is the 100th?
- 5Today is Friday. What day is it after 20 days?
- 6Today is Sunday. What day is it after 45 days?
- 7Which month is 25 months after January?
- 8Meena says the 22nd item of R G B Y is the 5th, from 22 ÷ 4 = 5. What is the mistake?
Show answers (try them yourself first)
- 1G
- 2R (25 ÷ 4 = 6 remainder 1)
- 3Y
- 4100 ÷ 3 = 33 remainder 1, so red.
- 5Thursday
- 6Wednesday
- 7February
- 8The remainder (2) gives the place, so it is G.
Special number patterns
Seeds on a sunflower and the number of petals follow special number patterns, and so do tiles in a square floor. Some number patterns have famous names.
Square numbers 1, 4, 9, 16, 25, 36. Triangular numbers 1, 3, 6, 10, 15, 21. Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21: each is the sum of the two before it.
In the figure : Square and triangular number dot pictures, and the Fibonacci rule.
Rule 1: Square and triangular numbers — Square: n × n. Triangular: 1 + 2 + ... + n.
- Easy : The 12th square number is 12 × 12 = 144.
- Medium : The 10th triangular number is 1 + 2 + ... + 10 = 55.
- Exam-type : The sum of the first 8 odd numbers is 8 × 8 = 64 (1 + 3 + ... + 15).
Rule 2: Fibonacci and pairing — Fibonacci: add the last two. Pairing: first + last for each pair.
- Easy : After 21 comes 13 + 21 = 34, then 21 + 34 = 55.
- Medium : 1 + 2 + ... + 20 = 10 pairs of 21 = 210.
- Exam-type : 1 + 2 + ... + 100 = 50 pairs of 101 = 5,050.
A rule with position n: 3, 7, 11, 15 follows 4n − 1; the 10th number is 4 × 10 − 1 = 39.
Starting Fibonacci with 1, 2 instead of 1, 1. Both starting numbers are 1.
Memory clue: Fibonacci: add the last two; Square: n times n.
Spot the difference
| Name | Numbers |
|---|---|
| Square | 1, 4, 9, 16, 25 |
| Triangular | 1, 3, 6, 10, 15 |
| Fibonacci | 1, 1, 2, 3, 5, 8 |
Practice questions
- 1Write the 12th square number.
- 2Write the 10th triangular number.
- 3Find the next two Fibonacci numbers after 21.
- 4Fill in: 1, 1, 2, 3, __, 8.
- 5Find 1 + 3 + 5 + 7 + 9.
- 6Find 1 + 2 + 3 + ... + 20 by pairing.
- 7Find 1 + 2 + 3 + ... + 100.
- 8Rule: 3, 7, 11, 15, ... follows 4n − 1. Find the 10th number and check by adding 4 repeatedly.
Show answers (try them yourself first)
- 1144
- 255
- 334, 55
- 45
- 525 (5 × 5)
- 610 pairs of 21 = 210
- 750 × 101 = 5,050
- 84 × 10 − 1 = 39. Check: 3, 7, 11, 15, 19, 23, 27, 31, 35, 39.