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MathematicsClass 6Patterns In

Patterns In | Class 6 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 3 min read

Patterns In – this guide gives you a concise, exam-ready overview of Patterns In from Class 6 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

1.4 Relations among Number Sequences

Number sequences are often related in surprising and beautiful ways. One classic example is the sum of odd numbers starting from 1:

1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 1 + 3 + 5 + 7 + 9 + 11 = 36

This shows that the sum of the first n odd numbers is n², a square number. This pattern holds forever.

A pictorial explanation helps understand why this happens. Square numbers can be represented by dots arranged in square grids. These grids can be partitioned into layers of dots corresponding to odd numbers: 1, 3, 5, 7, ... Each layer adds an odd number of dots, building up the square.

Similarly, adding counting numbers up and then down, such as 1, 1+2+1, 1+2+3+2+1, 1+2+3+4+3+2+1, ... also produces square numbers. This can also be explained pictorially by arranging dots in symmetric patterns.

These relationships show how sequences can be combined or related to produce new sequences, often with geometric interpretations.

Other examples include adding pairs of consecutive triangular numbers to get square numbers, sums of powers of 2, and relations involving hexagonal numbers and cubes.

Exploring these relations deepens understanding of number theory and pattern recognition.

📊 Diagram: Diagrams show square grids partitioned into layers representing odd numbers, and symmetric dot arrangements representing counting numbers added up and down.

🧪 Activity: Figure it Out: Students are asked to find pictorial explanations for these relations, calculate sums of large numbers of odd or counting numbers, explore sums of other sequences, and find their own patterns.

🔗 Connection: Introduces the idea of shape sequences and their relation to number sequences in the next section.

Frequently asked questions

Q.2. Why are 1, 3, 6, 10, 15, ... called triangular numbers? Why are 1, 4, 9, 16, 25, ... called square numbers or squares? Why are 1, 8, 27, 64, 125, ... called cubes?

These sequences are named based on the shapes formed by arranging dots:

  • Triangular numbers (1, 3, 6, 10, 15, ...) represent dots arranged in an equilateral triangle.
  • Square numbers (1, 4, 9, 16, 25, ...) represent dots arranged in a perfect square.
  • Cube numbers (1, 8, 27, 64, 125, ...) represent dots arranged in a perfect cube.

Refer to Table 2 on page 4 for pictorial illustrations.

Q.3. You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this!

36 is a triangular number because it can be arranged as a triangle with 8 rows (sum of first 8 natural numbers = 36). It is also a square number because 6 × 6 = 36, so dots can be arranged in a 6 by 6 square. Draw these arrangements in your notebook to visualize.

Q.4. What would you call the following sequence of numbers?

The answer given is 61. (Note: The original question seems incomplete or missing the sequence itself in the text.)

Q.5. Can you think of pictorial ways to visualise the sequence of powers of 2? powers of 3?

For powers of 2, refer to the sequence given on page 6. For powers of 3, one way is to visualize cubes growing in size, as shown in the images (img-12.jpeg to img-15.jpeg) illustrating the cubes of 1, 3, 9, 27, etc.

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