Sequences

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

a_n = 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

Find the next term of the following sequence.
Write down in words what it takes to go from the top line to the bottom line.
Find the general term.

{1, 4, 9, 16, 25, ...}
{2/3, 3/4, 4/5, 5/6, 6/7, ...}
{1, -1, 1, -1, 1, -1, 1, -1, ...}
{-1, 1, -1, 1, -1, 1, -1, 1, ...}
{3/2, -6/4, 9/8, -12/16, 15/32, ...}

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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