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:
Next notice what it takes to go from the top line to the bottom line. Each number must be multiplied by 2.
Exercises
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 a_{1} = 40,000 a_{2} = 40,000(1.03) = 41,200 a_{3} = 41,000(1.03) = 40,000(1.03)^{2} = 42,436 a_{4} = 40,000(1.03)^{3} = 43,709.08 a_{5} = 40,000(1.03)^{4} = 45,020.35
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