SEQUENCES OF REAL NUMBERS
Analysis
 
3. DEFINING A SEQUENCE OF REAL NUMBERS
A sequence is an infinite set of real numbers (the numbers which form the sequence are called terms).
All sequences have a first term and each term is followed by another.
Use the red and blue arrows to change the value of n or click on the animate button to see the sequence advance.

Use the Init button to restore the initial values.

4.- Look carefully at the sequence of fractions in the example.

You can see where each term is located in the sequence and their approximate decimal values by using the arrow keys at the bottom of the screen.

You can also write the value of n in the white box and click on Enter.

5.- In your exercise book write down the 1st, 10th, 100th, 1,000th, 10,000th and 12,345th terms together with the terms that come before and after each one. Is there a final term?
   
4. EXPRESSING ANY TERM IN A SEQUENCE OF REAL NUMBERS

This sequence can be denoted in a way which allows us to find the value of each term if we know its position in the sequence:

6.- Look at the following terms:

5th term = 1/5
12th term = 1/12
200th term = 1/200
1,789th term = 1/1789
nth term = 1/n

This last expression is called the:
General Term or nth term
and in this case is denoted by:

an= 1/n

which is read as:
a subscript n is equal to 1 divided by n

This means that any term can be obtained by dividing 1 by its position in the sequence.
 
5. THE GENERAL TERM OF A SEQUENCE OR NTH TERM (an)

When a sequence can be represented by an algebraic expression (an) which allows us to work out its terms, as is the case in the example above, this expression is known as its general term or nth term. Most sequences you are going to work with will have a general term.

7.- In your exercise book write down the general term or nth term of the sequence in this window, the first ten terms of the sequence and the 23rd, 289th, 1,578th and 25,784th terms as well as its approximate decimal value.

8.- Do you know how to obtain any term? Write down how you would do it in your exercise book together with three examples. 

9.- What would the last term in the sequence be?
       
           
  Juan Madrigal Muga
 
Spanish Ministry of Education. Year 2002
 
 

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