The iterative processes can be seen in our daily life. It can be seen in number sequences also. The numbers may increase or decrease following a pattern.

Iterative Process in Numbers

The above iterative processes can be seen in our daily life. It can be seen in number sequences also. The numbers may increase or decrease following a pattern.

1. Observe the following sequences and find the pattern that generates each one of them.

â€¢ 1, 3, 5, 7, â€¦ The pattern which generates these numbers is 1, 1+2, 3+2, 5+2...

â€¢ 50, 48, 46, 44,â€¦ The pattern which generates these numbers is 50, 50-2, 48-2, 46-2...

â€¢ 2, 4, 6,â€¦ The pattern which generates these numbers is 1Ã—2, 2Ã—2 ; 3Ã—2, ...

â€¢ 1, 4, 9, 16â€¦ The pattern which generates these numbers is 1Ã—1, 2Ã—2 , 3Ã—3...

â€¢ 2, 6, 12, 20, 30,â€¦ The pattern which generates these numbers is 1Ã—2, 2Ã—3, 3Ã—4, 4Ã—5, ...

â€¢ 2, 4, 8,16,â€¦ The pattern which generates these numbers is 2Ã—1, 2Ã—2, 2Ã—2Ã—2...

2. Observe the pattern, 1, 10, 100, ... . When the number of zeros increase the value also increases.

3. In the same way, can you guess the next number in the special number sequence given below?

1, 1, 2, 3, 5, 8, 13, 21, 34,â€¦

Yes it is 55, how? You have got it by adding 21 and 34. Havenâ€™t you?

Are you able to recognize the pattern in the above sequence? Yes, if we add the previous two consecutive terms, we get the next term as

1+1 =2, 1+2=3, 2+3=5, 3+5=8 , 5+8=13,â€¦

This special pattern of numbers is called the *Fibonacci sequence*. Each term in the Fibonacci sequence is called a *Fibonacci number*.

4.* Lucas numbers *form a sequence of numbers like the Fibonacci numbers and they* *are closely related to the Fibonacci numbers. Instead of starting with 1 and 1, Lucas numbers start with 1 and 3. The Lucas sequence is 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...

In all the above patterns of numbers, we can see the iterative processes.

Try these

i) Find the 10th term of the Fibonacci sequence.

**Answer: **55

ii) If the 11th term of the Fibonacci sequence is 89 and 13th term is 233 then, what is the 12th term?

**Answer: **144

Fibonacci numbers in nature

We come across the existence of Fibonacci sequence in many natural phenomena like spiral in a shell, arrangement of petals in flowers, branches of a tree, seeds in the head of a sunflower, petals on a daisy, the cells in the bee-hive, etc. Mathematical patterns are found in the distinct marking on animals and the structure of seashells also.

Note

We can also begin the Fibonacci sequence with 0 and 1, instead of 1 and 1.

Think

Are two consecutive Fibonacci numbers relatively prime?

DO YOU KNOW?

Golden Ratio:

Consider the ratio of successive Fibonacci numbers 3/2 = 1.5, 5/3 = 1.66, 8/5 = 1.6, 13/8 = 1.625, 21 /13 = 1.6153,â€¦) you can see the pattern, getting closer to 1.618 and that is denoted by Î¦ called the Golden Ratio (Î¦=1.618). It is observed that shapes having Golden Ratio appear beautiful.

DO YOU KNOW?

The Portrait of Mona Lisa has Fibonacci spiral pattern. This is one of the reasons for the enhanced beauty of Mona Lisa.

Tags : Information Processing | Term 3 Chapter 5 | 6th Maths , 6th Maths : Term 3 Unit 5 : Information Processing

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