SequencesA sequence, or progression, is a set of numbers obeying a simple rule. In a sequence, the successive terms (or numbers) are in definite order and formed according to some rule.
2, 4, 6, 8, 10, ... (1.0)
13, 18, 23, 28, 33, ... etc. (1.1)
Example (1.0) is the sequence of positive even integers. We denote
nth term of a general sequence by [tex]u_{n }[/tex], and the sequence by [tex]u_{1 }[/tex], [tex]u_{2 }[/tex], [tex]u_{3 }[/tex], [tex]u_{4 }[/tex], ..., [tex]u_{n }[/tex], [tex]u_{n+1 }[/tex], [tex]u_{n+2 }[/tex], ... . The simple rule defining a general sequence, [tex]u_{n }[/tex], is often given in the form of a function of
n, although this is not necessarily so. For examples (1.0) and (1.1), we have [tex]u_{n }[/tex]=2
n and [tex]u_{n }[/tex]=5
n+8.
Example 1: Find [tex]u_{n }[/tex] in terms of
n for the following sequences.
(i) 1, 3, 5, 7, 9, ...
(ii) 1, ⅛, 1/27, 1/64, ...
(iii) -1, 2, -3, 4, -5, ...
(iv) 1, 4, 9, 16, ...
(v) 1, -1, 1, -1, 1, ...
Solution:
(i) The terms in the sequence can be written as
1+0, 2+1, 3+2, 4+3, 5+4, ...
[tex]u_{n }[/tex]=
n+(
n-1)=2
n-1
(ii) The terms in the sequence can be written as
1/[tex]1^{3}[/tex], 1/[tex]2^{3}[/tex] ,1/[tex]3^{3}[/tex], 1/[tex]4^{3}[/tex], …
[tex]u_{n }[/tex]=1/[tex]n^{3}[/tex]
(iii) Inspecting this sequence, the terms can be written as
1⋅[tex](-1)^{1}[/tex], 2⋅[tex](-1)^{2}[/tex], 3⋅[tex](-1)^{3}[/tex], 4⋅[tex](-1)^{4}[/tex], …
[tex]u_{n }[/tex]=
n[tex](-1)^{n}[/tex]
Try (iv) and (v).
A good example of a sequence where [tex]u_{n }[/tex] cannot be written in form of a function of
n is the Fibonacci sequence;
[tex]u_{1 }[/tex]=0;
[tex]u_{2 }[/tex]=1;
[tex]u_{n }[/tex]=[tex]u_{n-1 }[/tex]+[tex]u_{n-2 }[/tex];
n≥3
In this type of sequence, there is mathematical relation between consecutive terms. This is called recurrence.
Example 2. Find the 4th term in the sequence with general term 3⋅[tex]5^{n-1}[/tex].
Solution:
The 4th term in the sequence with general term
3⋅[tex]5^{n-1}[/tex] is 3⋅[tex]5^{4-1}[/tex]=3⋅[tex]5^{3}[/tex]=3⋅125=375.
Read more:
Get Terms from Sequences