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 ?

Triangular Numbers The sum of the previous number and the order of the succeeding number results in the sequence of triangular numbers. 1 1 + 2 = 3 1 + 2 + 3 = 6 1 + 2 + 3 + 4 = 10 1 + 2 + 3 + 4 + 5 = 15 Triangular numbers are called so because these numbers of dots can be arranged to make an equilateral triangle. Thus, 1 , 3 , 6 , 10 , 15 , . . . are called triangular numbers. Square Numbers The numbers obtained by multiplying themselves in a sequence are called square numbers. 1 × 1 = 1 2 × 2 = 4 3 × 3 = 9 4 × 4 = 16 5 × 5 = 25 Square numbers are called so because these numbers of dots can be arranged to make a square. Thus, 1 , 4 , 9 , 16 , 25 , . . . . are called square numbers. Cubes The numbers obtained by multiplying themselves with their squares in a sequence are called cubes. 1 × 1 = 1 2 × 4 = 8 3 × 9 = 27 4 × 16 = 25 5 × 25 = 125 Cubes are called so because these many unit cubes can be arranged to make a cube. Thus, 1 , 8 , 27 , 64 , 125 , . . . . are called cubes.

What happens when you add up pairs of consecutive triangular numbers ? That is, take 1 + 3 , 3 + 6 , 6 + 10 , 10 + 15 , … . Which sequence do you get ? Why ? Can you explain it with a picture ?

1 + 3 = 4 = 2 × 2 = 2 2 ; 3 + 6 = 9 = 3 × 3 = 3 2 ; 6 + 10 = 16 = 4 × 4 = 4 2 The above pairs give square numbers.

What happens when you start to add up hexagonal numbers, i.e., take 1 , 1 + 7 , 1 + 7 + 19 , 1 + 7 + 19 + 37 , … ? Which sequence do you get ? Can you explain it using a picture of a cube ?

Hexagonal No. are: 1 , 7 , 19 , 37 , … … … . Let us add them: 1 = 1 3 ( third power of 1 ) 1 + 7 = 8 = 2 × 2 × 2 = 2 3 ( third power of 2 ) 1 + 7 + 19 = 27 = 3 × 3 × 3 = 3 3 ( third power of 3 ) 1 + 7 + 19 + 37 = 64 = 4 × 4 × 4 = 4 3 ( third power of 4 ) 1 + 7 + 19 + 37 + 61 = 125 = 5 × 5 × 5 = 5 3 ( third power of 5 )

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Earn 100

Copy the pictorial representations of the number sequences in Table $2$ in your notebook, and draw the next picture for each sequence!

Important Questions on Patterns in Mathematics

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Ganita Prakash Grade 6 Mathematics Textbook>Chapter 1 - Patterns in Mathematics>Q 5

Why are

1, 3, 6, 10, 15, \dots

called triangular numbers

?

Why are

1, 4, 9, 16, 25, \dots

called square numbers or squares

?

Why are

1, 8, 27, 64, 125, \dots

called cubes

?

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Ganita Prakash Grade 6 Mathematics Textbook>Chapter 1 - Patterns in Mathematics>Q 6

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! This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways!

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Ganita Prakash Grade 6 Mathematics Textbook>Chapter 1 - Patterns in Mathematics>Q 7

What would you call the following sequence of numbers $?$

Question Image

That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence $?$

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Ganita Prakash Grade 6 Mathematics Textbook>Chapter 1 - Patterns in Mathematics>Q 13

What happens when you add up pairs of consecutive triangular numbers

?

That is, take

1 + 3, 3 + 6, 6 + 10, 10 + 15, \dots .

Which sequence do you get

?

Why

?

Can you explain it with a picture

?

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Ganita Prakash Grade 6 Mathematics Textbook>Chapter 1 - Patterns in Mathematics>Q 15

What happens when you multiply the triangular numbers by

6

and add

1 ?

Which sequence do you get

?

Can you explain it with a picture

?

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Ganita Prakash Grade 6 Mathematics Textbook>Chapter 1 - Patterns in Mathematics>Q 16

What happens when you start to add up hexagonal numbers, i.e., take $1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, \dots ?$ Which sequence do you get $?$ Can you explain it using a picture of a cube $?$