Sequences

Definition of a Sequence

We define a sequence to be a function where the domain is the set of all positive integers and the range is the set of real numbers.  In other words, a sequence is an infinite list of real numbers.  

 

Example 

{2, 4, 6, 8, 10, 12, ...}

We use the notation  an  to indicate this function f(n) where n is a positive integer.  We can see this as the function

        an   =  2n

We can find this function as follows:  We write the number 1, 2, 3,... above the sequence:

1 2 3 4 5 6
2 4 6 8 10 12

Next notice what it takes to go from the top line to the bottom line.  Each number must be multiplied by 2.

 

Exercises

 

  1. Find the next term of the following sequence.

  2. Write down in words what it takes to go from the top line to the bottom line.

  3. Find the general term.

 

  1. {1, 4, 9, 16, 25, ...}
     36, square it, n^2

  2. {2/3, 3/4, 4/5, 5/6, 6/7, ...}
     7/8, add 1 for the numerator and 2 for the denominator, (n + 1)/(n + 2)

  3. {1, -1, 1, -1, 1, -1, 1, -1, ...}
     1, if even write -1 if odd write 1, (-1)^(n+1)

  4. {-1, 1, -1, 1, -1, 1, -1, 1, ...}
     -1, if odd write -1 if even write 1, (-1)^n

  5. {3/2, -6/4, 9/8, -12/16, 15/32, ...}
     -18/64, the sign as in #3 the numerator is times 3 and the denominator is 2 to the power, (-1)^(n+1)(3n)/(2^n)

 

Applications

Example

My starting salary was $40,000 per year.  Each year we receive a cost of living adjustment (COLA) of three percent of our original salary.  Write a sequence showing my salary for my first five years of working here.

Solution

We have

        a1  =  40,000

        a2  =  40,000(1.03)  =  41,200

        a3  =  41,000(1.03)  =  40,000(1.03)2  =  42,436

        a4  =  40,000(1.03)3  =  43,709.08

        a5  =  40,000(1.03)4  =  45,020.35

 

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