6.1 - Introduction to Sequences
Mathematically, a sequence is "a function whose domain is a subset of the natural numbers".
Examples:
i) 1, 3, 5, 7, 9... ii) 45, 22, 89, 63, 17... iii) 1000, 200, 50...
As seen in example (ii), a sequence does not necessarily follow a certain logical order. However, sequences of interest to study usually follow a particular pattern. Thus, simply put, a sequence is a set of numbers that shows a certain pattern.
- The values in the range of the function are referred to as terms
- The terms of a sequence are usually named using a single letter:
t1, t2, t3, t4,...tn
- "n" usually denotes the place of the term; ie. the "n"th term.
Example - find the first three terms:
(a) tn = 3n + 2
Solution:
(a) tn = 3n + 2
t1 = 3(1) + 2
= 5
t2 = 3(2) + 2
= 8
t3 = 3(3) + 2
= 11
Therefore, the first 3 terms are 5, 8 and 11.
6.2 Type I: Arithmetic Sequences
Each successive term is created by adding a constant number (the common difference).
Ex. 3, 7, 11, 15, 19...
Formula:
tn = a + (n - 1)d
Where tn is the term, a is the first term, d is the common difference and n is the number of the term (the "n"th term).
Example:
Find t10 for the following arithmetic sequence: 1, 3, 5, 7, ...
Solution:
Given a= 1 ; d = 3-1 = 2
t10 = 1 + (10-1)(2)
t10 = 19
Therefore, the 10th term of the sequence is 19.
Other examples may ask of you to determine a figure otherwise. To do so, simply sub in any given values into the formula.
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