2.11.1 Sequences
A sequence is an ordered list of numbers which follows a particular rule. Each number in a sequence is known as a term of the sequence.
The position of a term within a sequence is referred to as its location.
The letter \( n \) is frequently used for (unknown) position.
Subscript notation is used to refer to a specific term.
\( a_2 \) would be the second term in a sequence.
\( a_1 \) would be the \( 11^{th} \) term.
\( a_n \) would be the \( n^{th} \) term.
There are some useful rules to work with sequences.
Position-to-term rule:
A position-to-term rule gives the \( n^{th} \) term of a sequence in terms of \( n \).
The link between \( n \) and \( a_n \) is that \( a_n \) is always \( 3 \) more than \( n \).
So, the position-to-term rule is
\( a_n =n+3 \)
Term-to-term rule:
A term-to-term rule gives the \( (n+1)^{th} \) term in terms of the \( n^{th} \) term.
The difference between any two consecutive terms is \( 1 \).
So, the term-to-term rule is
\( a_{n+1} – a_n = 1 \) or \( a_{n+1}=a_n + 1 \)
Types of sequences:
There are four main types of different sequences.
Arithmetic sequences
An arithmetic sequence is an ordered collection of numbers with a common difference between them.
e.g.
\( 4, 7, 10, 13, 16,… \)
Geometric sequences
A geometric sequence is a set of ordered numbers that progress by multiplying or dividing each term by a common ratio.
e.g.
\( 1, 2, 4, 8, 16, … \)
Quadratic sequences
A quadratic sequence is an ordered set of numbers that follow a sequence-based rule \( n^2=1, 4, 9, 16, 25, … \)
The difference between each term is not equal but the second difference is.
e.g.
\( 4, 7, 12, 19, 28, …. \)
Special sequences
You must be able to recognize some critical special sequences.
Fibonacci sequence:
The Fibonacci sequence is a series of integers that begin with a zero, then a one, another one, and a series of steadily increasing numbers. The sequence follows the rule that each number is equal to the sum of the two numbers before it.
\( 0, 1, 1, 2, 3, 5, 8, 13, 21, … \)
Fractional sequences:
\( 1,\frac{1}{2},\frac{1}{3},\frac{1}{4}, … \)
Such sequences do not fall into any category but the link between \( n \) and \( a_n \) is fairly easy to spot.
Identification of sequences:
Sometimes it’s obvious but if not then check
If there is a number that you multiply to get from one term to the next then it’s a geometric sequence.
If the first order difference is constant between the terms, and is known as a common difference denoted by d, then it’s a linear sequence.
If the second order difference is constant between the terms then it’s a quadratic sequence.
Worked example:
A sequence has \( n^{th} \) term \( 2n^2+5n-15 \).
Find the difference between the \( 4^{th} \) term and the \( 5^{th} \) term of this sequence.
Substitute \( n = 4 \) in \( 2n^2+5n-15 \) to get \( 4^{th} \) term
\( 2(4)^2+5(4)-15 \)
\( =32+20-15 \)
\( =37 \)
Substitute \( n=5 \) in \( 2n^2+5n-15 \) to get \( 5^{th} \) term
\( 2(5)^2+5(5)-15 \)
\( =50+25-15 \)
\( =60 \)
Difference between the \( 4^{th} \) and \( 5^{th} \) term is
\( 60-37=23 \)
\( 23 \)