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Arithmetic Sequences

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ARITHMETIC SEQUENCES<BR>

ARITHMETIC SEQUENCES


A child first becomes aware of numbers through counting.
Arranged in order, the counting numbers form a number sequence:

    1 2 3 4 5 6 7 8 9 10 11 ...

A number sequence is an arrangement of numbers in which each
successive number follows the last according to a uniform rule.

For the sequence of counting numbers, the rule is “add 1 to each number to get the next number.” The symbol “. . .”, called an ellipsis, indicates that the sequence continues. A mathematician once said: “A number sequence is like a bus; nobody ever doubts that there is always room for one more.”

The numbers in a sequence are called its terms.
To represent the terms of a number sequence, we will use the notation

    t1 t2 t3 t4 t5 . . . tn.

The little number on the right of each t is called a subscript and is the term number, or index.
It keeps track of where we are in the sequence;
t3 (pronounced "tee-sub-three”), for example, represents the third term.
The symbol tn represents the nth term.

Each successive term of the sequence 8 20 32 44 56 ...

can be found by adding the same number to the preceding term.

Such a sequence is called arithmetic (pronounced “arithmetic”).

An arithmetic sequence is a number sequence in which each successive term may be found by adding the same number.

Another example of an arithmetic sequence is illustrated by this figure.

Poker Chips

The first term is 2 (t1 = 2) and the number that is added is 3.
This number is also called the common difference of the sequence
because it is the difference between each pair of successive terms:

    5-2=3, 8-5=3, 11-8=3,

and so on...



QUESTIONS



    The figure below shows three stacks of poker chips.
    The stacks illustrate the first three terms of an arithmetic sequence.


    Poker Chips


  1. What are the terms?

  2. What is their common difference?

  3. What are the next three terms of the sequence?



    The 100th term of the arithmetic sequence

        1 2 3 4 5 ...

    is obvious: it is 100. The 100th term of the arithmetic sequence

        2 5 8 11 14 ...

    however, is not obvious at all.
    One way to find out what it is would be to continue writing the sequence until we arrive at it.
    There is an easier way, however.
    Poker Chips

    Look at the diagram above and the accompanying pattern below.

    
    The 1st term, t1, is    2 + 3 x 0 = 2
    
    The 2nd term, t2, is    2 + 3 x 1 = 5
    
    The 3rd term, t3, is    2 + 3 x 2 = 8
    
    The 4th term, t4, is    2 + 3 x 3 = 11
    
    The 5th term, t5, is    2 + 3 x 4 = 14
    The 100th term must be

        2 + 3 x 99 = 2 + 297 = 299



  4. Find the 10th term of the arithmetic sequence

        3 10 17 24 ...

    by writing down the next six terms.

  5. Find the 10th term of the same sequence by using the shortcut suggested by the pattern:

    
The lst term, t1, is 3 + 7 x 0 = 3
    
The 2nd term, t2, is 3 + 7 x 1 = 10
    
The 3rd term, t3, is 3 + 7 x 2 = 17
    
The 4th term, t4, is 3 + 7 x 3 = 24


    Use shortcuts to find the indicated terms of the following arithmetic sequences.

  6. The 11th term of 8 15 22 29 ...

  7. The 25th term of 6 10 14 18 ...

  8. The 16th term of 100 97 94 91 ...

  9. The 40th term of 24 35 46 57 ...

  10. The 71st term of 5 30 55 80 ...



    The diagram below represents an arithmetic sequence whose first term is t1,
    and whose common difference is d.
    Expressions for the first three terms are t1, t1 + d, and t1 + 2d, respectively.
    Write expressions for the 4th and 5th terms, t4 and t5.
    Notice the relationship between the index n of the term and its number of differences d.
    
1st term: t1

2nd term: t1 + d

3rd term: t1 + d + d = t1 + 2d

4th term: t1 + d + d + d = ?

5th term: t1 + d + d + d + d = ?
  11. the 4th and 5th terms, t4 and t5 are _____ and _____.

  12. the 10th term, t10 is _____.

  13. the nth term, tn is _____.

  14. the nth + 1 term, tn+1 is _____.

  15. Add d to the nth term, tn, and simplify it.

  16. When you simplify it, is it equivalent to the nth + 1 term, tn+1?

  17. If it is equivalent, did you prove the nth + 1 term is true for all integers? Why?

  18. If it is NOT equivalent, what does that prove about the formula?

  19. What method of reasoning did you use to answer the previous two questions? Inductive or Deductive?

  20. What method of reasoning did you use to find the nth term? Inductive or Deductive?
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Last Modified 12/21/08 10:01 AM