This is a free lesson from our course in Algebra II
In this lesson you'll walk through some basic terminology associated with sequences.
Then you'll be going over to basics of special forms i.e.
• Arithmetic sequences.
• Geometric sequences. All this will be done using some examples with
solution and with the help of audio, video presentation in instructor's own hand
writing. A sequence is a set of numbers written in a given order. A finite sequence
is a function whose domain is the set of integers and an infinite sequence is
a function whose domain is the set of positive integers.(More text below video...)
(Continued from above) By a sequence, we mean an arrangement of numbers in a definite order according
to some rule. We denote the terms of a sequence by a_{1}, a_{2}, a_{3}, ... , etc., the subscript denotes the position of the term. Sequences can have a certain "rule" by which terms progress, but they can also be completely random. {1, 2, 3, 4 ,...} is a very simple sequence (and it is an infinite sequence) {1, 3, 5, 7} is the sequence of the first 4 odd numbers (and is a finite sequence). A
rule that makes work for any term of a sequence can often be determined from the
first few terms of the sequence.
E.g. the next three terms of the sequence 2, 4, 8, 16,... are: 32, 64, 128.
An arithmetic sequence is a sequence of numbers such that the difference of any
two successive numbers is a constant. Any term of an arithmetic sequence can be
evaluated with the formula:
a_{n}
=a_{1} +(n  1)d,
where 'd ' is the common difference of the sequence.
E.g. The sequence 2, 5, 8, 11, 14... is an arithmetic sequence with common difference
3.
A geometric sequence is a sequence such that for all n, there is a constant
r such that
a_{n}/a_{n}_{1}
= r.
The constant r is called the common ratio. E.g. the 10th term of the sequence
4, 12, 36, 108, 324,... is 78,732.
Now we will introduce and go through some basic terminology associated with series
and include working on arithmetic and geometric series in the sections following
this. Summation (
)
notation is a shorthand way of saying take the sum of certain terms of a sequence.
The sum of terms 1 through n will be expressed as:
(i = 1 to n)a_{i}=a_{1}
+ a_{2} + a_{3} +...+a_{n}
For example,
Sum of n =
1 to n = 4 is 24.
A series is a sum of terms of a sequence and it can be written in summation notation:
(i=4 to 7)3i
or expanded out in a sum 12 + 15 + 18 +21.
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