Relations among Number Sequences

Class 6 Maths · Chapter 1: Patterns in Mathematics · NCERT Ganita Prakash section 1.4

Learn

Start with 1 square.

1 = 1 × 1

Add an L-shape of 3 squares around it. Now it is a 2 × 2 square.

1 + 3 = 4 = 2 × 2

Add an L of 5. Now it is 3 × 3.

1 + 3 + 5 = 9 = 3 × 3

Try it yourself: add layers and watch the square grow.

Each new L is 2 bigger, so the L's are the odd numbers. Adding the first n odd numbers always makes an n × n square.

1 + 3 + 5 + 7 + 9 = 25 = 5 × 5

Another pattern: count up and down. 1 + 2 + 3 + 4 + 3 + 2 + 1. Turn the dots on their side and they fit into a 4 × 4 square.

1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4 × 4

Add two triangular numbers next to each other: the blue triangle (15) and the coral triangle (10) fit together into a square.

10 + 15 = 25 = 5 × 5

Add up powers of 2: 1, 1 + 2, 1 + 2 + 4 … The total is always one less than the next power of 2. The dashed box is the missing 1.

1 + 2 + 4 + 8 = 15 = 16 − 1

Add up hexagonal numbers and you get cube numbers! Each hexagonal number is one visible shell of a cube.

1 + 7 + 19 + 37 = 64 = 4 × 4 × 4

Remember

• Adding the first n odd numbers gives n × n.
• 1 + 2 + … + n + … + 2 + 1 = n × n.
• Two triangular numbers next to each other add up to a square number.
• 1 + 2 + 4 + 8 + … is one less than the next power of 2.

Quick check

1 + 3 + 5 + 7 + 9 + 11 + 13 = ?

1. 42
2. 45
3. 49
4. 56

Think about it

What do the first 12 odd numbers add up to? Can you find it without adding them one by one?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.