Number Sequences and Patterns
Number sequences and patterns is the science of numbers. Our mathematical initiation starts with the number sequence 1, 3, 4, . . . .At various stages of our mathematics education, we get acquainted with various number sequences, such as the even numbers 2, 4, 6, … , the odd number 1, 3, 5, 7 …, the perfect squares are 1, 4, 9, …, and so on.
Number patterns
Suppose we write the patterns of numbers as
2, 4, 6, 8, 10 …
What number do we write next patterns?
What about
2, 4, 8, 16, 32 …?
and
1, `(1)/(3)` , `(1)/(5)` , `(1)/(7)` , `(1)/(9)` …?
Such a patterns of numbers formed according to some rule are called number sequences.
Look at this sequences of numbers:
5, 9, 13, 17, 21 …
What is the next number? 25, isn’t it? How did you find it? How do we get the numbers in this sequences?
5 + 4 = 9
9 + 4 = 13
13 + 4 = 17
17 + 4 = 21
and so on.
Now look at
2, 7, 12, 17, 22 …
How do we get the numbers of this sequences? We start from 2 and repeatedly add 5. Sequences like these are called arithmetic sequences or more often arithmetic progressions. The numbers belonging to a sequence are called terms of the sequence.
Thus for example, in the arithmetic progression
2, 7, 12, 17, 22…
The first term is 2, the second term is 7, and the third term is 12 and so on.
Different types of sequences and patterns
Look at the sequence of pattern below:
1, `(3)/(2)` , 2, `(5)/(2)` , 3, `(7)/(2)` …
Is this an arithmetic progression? Look at its term once again:
The sequences are,
1 + `(1)/(2)` =`(3)/(2)`
`(3)/(2)` +`(1)/(2)` = 2
2 + `(1)/(2)` = `(5)/(2)`
`(5)/(2)` + `(1)/(2)` = 3
and so on.
Thus the terms of this sequence are got by starting from 1 and repeatedly adding ½. So, this is also an arithmetic progression.
Look at another one:
20, 15, 10, 5 …
Is this an arithmetic progression? Here, the terms decrease by 5 as
20- 5 = 15
15 – 5 = 10
10 – 5 = 5
and so on.
But can’t we write 20 - 5 as 20 + (-5)? Thus we have
20 + (-5) = 15
15 + (-5) = 10
10 + (-5) = 5
as the terms of the sequence. In other words we start from 20 and repeatedly add -5. So, this too is an arithmetic progression.
Number patterns
Suppose we write the patterns of numbers as
2, 4, 6, 8, 10 …
What number do we write next patterns?
What about
2, 4, 8, 16, 32 …?
and
1, `(1)/(3)` , `(1)/(5)` , `(1)/(7)` , `(1)/(9)` …?
Such a patterns of numbers formed according to some rule are called number sequences.
Look at this sequences of numbers:
5, 9, 13, 17, 21 …
What is the next number? 25, isn’t it? How did you find it? How do we get the numbers in this sequences?
5 + 4 = 9
9 + 4 = 13
13 + 4 = 17
17 + 4 = 21
and so on.
Now look at
2, 7, 12, 17, 22 …
How do we get the numbers of this sequences? We start from 2 and repeatedly add 5. Sequences like these are called arithmetic sequences or more often arithmetic progressions. The numbers belonging to a sequence are called terms of the sequence.
Thus for example, in the arithmetic progression
2, 7, 12, 17, 22…
The first term is 2, the second term is 7, and the third term is 12 and so on.
Different types of sequences and patterns
Look at the sequence of pattern below:
1, `(3)/(2)` , 2, `(5)/(2)` , 3, `(7)/(2)` …
Is this an arithmetic progression? Look at its term once again:
The sequences are,
1 + `(1)/(2)` =`(3)/(2)`
`(3)/(2)` +`(1)/(2)` = 2
2 + `(1)/(2)` = `(5)/(2)`
`(5)/(2)` + `(1)/(2)` = 3
and so on.
Thus the terms of this sequence are got by starting from 1 and repeatedly adding ½. So, this is also an arithmetic progression.
Look at another one:
20, 15, 10, 5 …
Is this an arithmetic progression? Here, the terms decrease by 5 as
20- 5 = 15
15 – 5 = 10
10 – 5 = 5
and so on.
But can’t we write 20 - 5 as 20 + (-5)? Thus we have
20 + (-5) = 15
15 + (-5) = 10
10 + (-5) = 5
as the terms of the sequence. In other words we start from 20 and repeatedly add -5. So, this too is an arithmetic progression.