Page 14 - Revised Maths Wisdom Class - 6

12 MATHS

Hence, we can make a sequence of square numbers 1, 4, 9, 16, 25, 36, ............. . This sequence can be extended
infinitely. We may also find some more interesting facts and patterns within the square numbers.

1 × 1 = 1 = 1
2
2 × 2 = 2 = 4 = 1 + 3 [Sum of the first two odd numbers]
2
3 × 3 = 3 = 9 = 1 + 3 + 5 [Sum of the first three odd numbers]
2
4 × 4 = 4 = 16 = 1 + 3 + 5 + 7 [Sum of the first four odd numbers]
2
5 × 5 = 5 = 25 = 1 + 3 + 5 + 7 + 9 [Sum of the first five odd numbers]
2
If you observe the above pattern carefully, each square number is the sum of consecutive odd numbers. Can you
guess the next step in the pattern?

5. Patterns in Triangular Numbers
Observe the following shapes made by dots.

Each of the given shapes is a triangle. Count the numbers of dots. These are 1, 3, 6, 10, 15,.... The numbers that
form a triangle are called triangular numbers.

Looking at the triangular numbers again and notice the difference 1st 2nd 3rd 4th 5th 6th
between two terms. 1 3 6 10 15 21
+ 2 + 3 + 4 + 5 + 6

You can extend the pattern easily. Hence the seventh and eighth triangular number will be
1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 and 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36.
In the same way, we can find 48th triangular number and so on.

6. Fibonacci Sequence
1 1
The Fibonacci sequence is a sequence of numbers 1 1 1
2 1 2 1
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... 3 1 1 4 3 6 3 4 1 1
The next number is found by adding up the two numbers 8 5 1 5 10 10 5 1
15
before it. 13 1 6 15 20 35 21 6 1 1
21 1 7 21 35 7 1
56 70
These numbers are known as Fibonacci numbers. 34 1 8 28 84 126 126 56 28 8 1
55 1 9 36 84 36 9 1
1 10 45 120 210 252 210 120 45 10
7. Cube Numbers 89
Cube numbers are formed by multiplying a number by itself, three times.

The first four cube numbers are as follows:
1 × 1 × 1 = 1 2 × 2 × 2 = 8 3 × 3 × 3 = 27 4 × 4 × 4 = 64