02) Number Patterns

 

Purpose of this lesson

After learning this lesson you should be able to do the following things.

1. To know how number patterns are formed
2. To get the ability to identify the general term of a number pattern
3. To identify any number in a sequence using the general term
4. To use the knowledge of number patterns and make some practical situations easier

First think for a while of what the term “Number patterns” mean. Certainly you know many examples for patterns. For example if you consider the handwriting of a set of students from a class, there will be a huge difference. One person may scribble on the book, while the other’s handwriting seems as neat as a printed material. Since this example is visual, we are able to understand it easily.

Some examples for number patterns

1, 3, 5, 7, 9, 11, 13, 15            -           Odd Numbers
2, 4, 6, 8, 10, 12, 14, 16          -           Even Numbers
1, 3, 6, 10, 15, 21, 28, 36        -           Triangular Numbers
1, 4, 9, 16, 25, 36, 49, 64        -           Square Numbers

All these are some examples for number patterns. Why do you want to learn this? Is this going to help us in any way?

An amazing number pattern

It helps us in some simple daily calculations and for estimations. By that you can take the example of simple interest for a bank loan. Don’t be too much confused at this point. I recommend you to get a text book and read the questions at the end of the chapter. You will be presented with a number of questions which relate your daily life situations. So this is how Math is going to help you. When you start to approach a question; first, free your mind. You should not have the thought of confusion whether you’ll be able to solve it or not, in turn you should consider it as a Real Life situation and add to your knowledge base.

In this lesson you are about to learn one type of number pattern as in our 1st and 2nd example above. They are the Odd and Even number sequence. What do you think is the common relationship in those patterns?
The difference between any two successive (for an example 3 and 5 are successive in the odd number series) terms is equal.
The examples are further clarified below.

Successive terms in the Odd number series


1 & 3
3 & 5
5 & 7
7 & 9
9 & 11

Successive terms in the Even number series


2 & 4
4 & 6
6 & 8
8 & 10
10 & 12

When you calculate the difference you should always subtract the smaller value from the bigger one in this example. (For number patterns)
Let’s calculate the difference between successive terms in the Odd number series.

3 – 1 = 2
5 – 3 = 2
7 – 5 = 2
9 – 7 = 2
11 – 9 = 2

The following is another example.

3, 11, 19, 27, 35, 43, 51


Our objective is to find the general term for the pattern above.

Step 1: Identify the first term (It’s not going to be a tough task – in the above example it is 3)
Step 2: Identify the common difference.
            11 – 3 = 8
            19 – 11 = 8
            27 – 19 = 8
            35 – 27 = 8
            43 – 35 = 8
            51 – 43 = 8
As you can see the common difference is equal in all situations. I’ve done all possible ways for the purpose of explanation. But it is sufficient to use any two successive terms to find the common difference.


The Equation for General Term is as follows. (Also called nth term)

General Term = First term + (common difference) x (n-1)

The general equation for the example above is

3(First term) + 8(n-1)

Now you should be able to find out the general term for any number pattern of the above type.
The next part is to use the General Term and find out any term in the series/pattern.

If you need find out the 2nd term you should substitute “two” in place of “n” in the general equation. Let’s find out the first five terms for the general term we derived above. So we can check whether we‘ve done our Math right or not.


1st term = 3+8*0 = 3 (The notation “*” means multiplication symbol.)
2nd term = 3+8*1 = 3+ 8 = 11
3rd term = 3+8*2 = 3 + 16 = 19
4th term = 3+8*3 = 3+ 24 = 27
5th term = 3+8*4 = 3+ 32 = 35

You can see that the number values match. So we performed our math correctly.
What if you were said to find out the 101st term? Simply substitute “100” in place of “n”.

101st term = 3 + (101-1)*8 = 3 + 100*8 = 3 + 800 = 803

That’s it for this lesson. Practice these ideas by applying it and solving some problems. If you want to learn about Scientific Notation click here.

Hope to see you seen. If you liked it or have any suggestions please make a comment below.