Class 6 Chapter 1 Patterns in Mathematics Worksheet for NCERT Book

In this chapter 1, Patterns in Mathematics mathematics worksheet, we will be solving “patterns”. We are going to spot patterns. We will also try to explain why they happen. This worksheet is going to give you practice with number sequences, relationships between sequences, and simple shape-pattern reasoning.

Class 6 Mathematics – Patterns in Mathematics | Practice Worksheet

A. Number Sequences: Find the Rule

1. Write the next three terms and the rule:
   1. 1, 3, 5, 7, 9, …
   2. 2, 4, 6, 8, 10, …
   3. 1, 4, 9, 16, 25, …
   4. 1, 8, 27, 64, 125, …
   5. 1, 2, 4, 8, 16, …
   6. 1, 3, 9, 27, 81, …
2. A sequence is: 1, 3, 6, 10, 15, 21, …
   1. Write the next three terms.
   2. Write how much each term increases by.
   3. What is the name of this sequence?
3. Virahāṅka numbers start as: 1, 2, 3, 5, 8, 13, 21, …
   1. Write the next three terms.
   2. Explain the rule in one line.
4. Identify which sequence each belongs to (Odd / Even / Squares / Cubes / Powers of 2 / Powers of 3 / Triangular):
   1. 64
   2. 36
   3. 27
   4. 21
   5. 49
   6. 32
5. Create your own sequence with rule:
   “Multiply by 2, then add 1”.
   Start at 1 and write first 6 terms.

B. Patterns That “Talk” to Each Other (Relations Among Sequences)

6. Compute the sums and observe the pattern:
   1. 1
   2. 1 + 3
   3. 1 + 3 + 5
   4. 1 + 3 + 5 + 7
   5. 1 + 3 + 5 + 7 + 9

   Then answer:

   1. What type of numbers do these sums form?
   2. Predict the sum of the first 10 odd numbers without adding all of them one by one.
   3. Predict the sum of the first 50 odd numbers.
7. “Up and down” pattern:
   1. 1
   2. 1 + 2 + 1
   3. 1 + 2 + 3 + 2 + 1
   4. 1 + 2 + 3 + 4 + 3 + 2 + 1

   Answer:

   1. Compute each value.
   2. What sequence do these values match?
   3. Find the value of 1 + 2 + 3 + … + 20 + 19 + … + 3 + 2 + 1.
8. Consecutive triangular numbers:
   Add the pairs and write the first five results:
   1. 1 + 3
   2. 3 + 6
   3. 6 + 10
   4. 10 + 15
   5. 15 + 21

   Which famous sequence do you get?

9. Powers of 2 “magic”:
   Start adding powers of 2:
   1. 1
   2. 1 + 2
   3. 1 + 2 + 4
   4. 1 + 2 + 4 + 8
   5. 1 + 2 + 4 + 8 + 16

   Now add 1 to each of these sums.
   What sequence do you get?

10. Hexagonal numbers begin: 1, 7, 19, 37, …
    1. Find the next term (use “difference pattern”).
    2. Now add them step-by-step: 1, 1+7, 1+7+19, 1+7+19+37.
    3. Which famous sequence appears in these totals?

C. Shape Patterns Without Drawing (Reasoning Only)

11. Regular polygons have 3 sides, 4 sides, 5 sides, 6 sides, …
    1. Write the next four names after hexagon.
    2. In a closed polygon, compare the number of sides and corners. What do you notice?
12. A “stacked squares” pattern has 1 small square in Step 1, 4 in Step 2, 9 in Step 3, 16 in Step 4.
    1. How many small squares in Step 7?
    2. Write the rule in words.
13. A “stacked triangles” pattern has 1 triangle in Step 1, 4 in Step 2, 9 in Step 3, 16 in Step 4.
    1. Which number sequence is this?
    2. Why might a triangle pattern produce square numbers? Write one possible reason.
14. Koch snowflake segment counts follow: 3, 12, 48, 192, …
    1. What is the multiplication factor each time?
    2. Find the next two terms.
    3. Write a rule using “powers of 4”.

Answer Key with Hints

A. Number Sequences: Find the Rule

1. Answers:
   1. Odd: 11, 13, 15. Rule: add 2.
   2. Even: 12, 14, 16. Rule: add 2.
   3. Squares: 36, 49, 64. Rule: n².
   4. Cubes: 216, 343, 512. Rule: n³.
   5. Powers of 2: 32, 64, 128. Rule: multiply by 2.
   6. Powers of 3: 243, 729, 2187. Rule: multiply by 3.

   Hint: Check whether the difference stays constant (odd/even), or whether the term is “a square/cube”, or whether each term is multiplied by the same number.

2. Answers:
   1. Next: 28, 36, 45.
   2. Increases: +2, +3, +4, +5, …
   3. Name: Triangular numbers.

   Hint: Triangular numbers grow by the counting numbers: +2, +3, +4, +5, …

3. Answers:
   1. Next: 34, 55, 89.
   2. Rule: each term = sum of previous two terms.

   Hint: 13 comes from 5+8, so continue the same idea.

4. Answers (one correct set):
   1. 64: Squares and also Powers of 2.
   2. 36: Squares and also Triangular (because 36 can be arranged as triangle and square).
   3. 27: Cubes and Powers of 3.
   4. 21: Triangular.
   5. 49: Squares.
   6. 32: Powers of 2.

   Hint: Some numbers can belong to more than one sequence.

5. Answer: 1, 3, 7, 15, 31, 63
   Hint: Each step: ×2 then +1.

B. Patterns That “Talk” to Each Other

6. Answers:
   1. 1 = 1
   2. 1+3 = 4
   3. 1+3+5 = 9
   4. 1+3+5+7 = 16
   5. 1+3+5+7+9 = 25

   Conclusions:

   1. These are square numbers.
   2. First 10 odd numbers sum = 10² = 100.
   3. First 50 odd numbers sum = 50² = 2500.

   Hint: Adding odd numbers starting from 1 gives squares.

7. Answers:
   1. 1 = 1
   2. 1+2+1 = 4
   3. 1+2+3+2+1 = 9
   4. 1+2+3+4+3+2+1 = 16
   5. So it matches square numbers.
   6. 1+2+…+20+19+…+2+1 = 20² = 400.

   Hint: The “up and down” sum forms a symmetric square pattern, so the result is n².

8. Answers:
   1. 1+3 = 4
   2. 3+6 = 9
   3. 6+10 = 16
   4. 10+15 = 25
   5. 15+21 = 36

   Conclusion: Square numbers.
   Hint: Adding consecutive triangular numbers gives squares.

9. Answers:
   1. 1
   2. 3
   3. 7
   4. 15
   5. 31

   After adding 1: 2, 4, 8, 16, 32 (powers of 2).
   Hint: The running sum of powers of 2 is “one less than the next power of 2”.

10. Answers:
    1. Differences: 6, 12, 18, … so next difference is 24, next term = 37+24 = 61.
    2. Totals: 1, 8, 27, 64.
    3. These totals are cube numbers.

    Hint: The chapter notes that adding hexagonal numbers gives cubes.

C. Shape Patterns Without Drawing

11. Answers:
    1. After hexagon: heptagon, octagon, nonagon, decagon.
    2. Number of sides = number of corners (vertices).

    Hint: Every side meets at a corner, so they match for closed polygons.

12. Answers:
    1. Step 7 = 7² = 49 small squares.
    2. Rule: Step n has n² squares.

    Hint: 1, 4, 9, 16 are perfect squares.

13. Answers:
    1. Square number sequence.
    2. One reason: the triangle can be built row-by-row as 1, then 1+2+1, then 1+2+3+2+1, which matches the “up and down” square pattern.

    Hint: The chapter connects stacked triangles with the “adding up and down gives squares” idea.

14. Answers:
    1. Multiply by 4 each time.
    2. Next terms: 768, 3072.
    3. Rule: 3 × 4ⁿ (starting with n=0 gives 3).

    Hint: Each step replaces every segment in a way that makes the total segments quadruple.