Arithmetic sequences
An arithmetic sequence is a sequence where consecutive terms are calculated by adding a constant value (positive or negative) to the previous term. We call this constant value the common difference (d
).
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For example,
3;0;−3;−6;−9;…
This is an arithmetic sequence because we add −3
to each term to get the next term:
The general term for an arithmetic sequence (EMCDQ)
For a general arithmetic sequence with first term a
and a common difference d
, we can generate the following terms:
T1T2T3T4⋮Tn=a=T1+d=a+d=T2+d=(a+d)+d=a+2d=T3+d=(a+2d)+d=a+3d⋮⋮⋮=Tn−1+d=(a+(n−2)d)+d=a+(n−1)d
Therefore, the general formula for the n
th
term of an arithmetic sequence is:
Tn=a+(n−1)d
 Arithmetic sequence
 An arithmetic (or linear) sequence is an ordered set of numbers (called terms) in which each new term is calculated by adding a constant value to the previous term:
Tn=a+(n−1)d
where
is the nth
 term;
 n
 is the position of the term in the sequence;
 a
 is the first term;
 d

 is the common difference.
Test for an arithmetic sequence
To test whether a sequence is an arithmetic sequence or not, check if the difference between any two consecutive terms is constant:
d=T2−T1=T3−T2=…=Tn−Tn−1
If this is not true, then the sequence is not an arithmetic sequence.
Worked example 1: Arithmetic sequence
Given the sequence −15;−11;−7;…173
.
 Is this an arithmetic sequence?
 Find the formula of the general term.
 Determine the number of terms in the sequence.
Check if there is a common difference between successive terms
T2−T1T3−T2∴This is an =−11−(−15)=4=−7−(−11)=4arithmetic sequence with d=4
Determine the formula for the general term
Write down the formula and the known values:
Tn=a+(n−1)d
a=−15;d=4
Tn=a+(n−1)d=−15+(n−1)(4)=−15+4n−4=4n−19
A graph was not required for this question but it has been included to show that the points of the arithmetic sequence lie in a straight line.
Note: The numbers of the sequence are natural numbers (n∈{1;2;3;…})
and therefore we should not connect the plotted points. In the diagram above, a dotted line has been used to show that the graph of the sequence lies on a straight line.
Determine the number of terms in the sequence
Tn173192∴n∴T48=a+(n−1)d=4n−19=4n=1924=48=173
Write the final answer
Therefore, there are 48
Arithmetic mean
The arithmetic mean between two numbers is the number halfway between the two numbers. In other words, it is the average of the two numbers. The arithmetic mean and the two terms form an arithmetic sequence.
For example, the arithmetic mean between 7
and 17
is calculated:
Arithmetic mean ∴7;12;17T2−T1T3−T2=7+172=12 is an arithmetic sequence=12−7=5=17−12=5
Plotting a graph of the terms of a sequence sometimes helps in determining the type of sequence involved. For an arithmetic sequence, plotting Tn
vs. n
results in the following graph:
 If the sequence is arithmetic, the plotted points will lie in a straight line.
 Arithmetic sequences are also called linear sequences, where the common difference (d
 ) is the gradient of the straight line.
Tncan be written as Tnwhich is of the =a+(n−1)d=d(n−1)+asame form as y=mx+c
 Quadratic sequence
 A quadratic sequence is a sequence of numbers in which the second difference between any two consecutive terms is constant.
The general formula for the nth
term of a quadratic sequence is:
Tn=an2+bn+c
It is important to note that the first differences of a quadratic sequence form an arithmetic sequence. This sequence has a common difference of 2a
between consecutive terms. In other words, a linear sequence results from taking the first differences of a quadratic sequence.
Worked example 2: Quadratic sequence
Consider the pattern of white and blue blocks in the diagram below.
 Determine the sequence formed by the white blocks (w)
 .
 Find the sequence formed by the blue blocks (b)
 .
Use the diagram to complete the table for the white blocks
We see that the next term in the sequence is obtained by adding 4
to the previous term, therefore the sequence is linear and the common difference (d) is 4.
The general term is:
Tn=a+(n−1)d=4+(n−1)(4)=4+4n−4=4n
Use the diagram to complete the table for the blue blocks
We notice that there is no common difference between successive terms. However, there is a pattern and on further investigation we see that this is in fact a quadratic sequence:
Tn=(n−1)2
Draw a graph of Tn
vs. n
for each sequence
White blocks: TnBlue blocks: Tn=4n=(n−1)2=n2−2n+1
Since the numbers of the sequences are natural numbers (n∈{1;2;3;…})
, we should not connect the plotted points. In the diagram above, a dotted line has been used to show that the graph of the sequence formed by the white blocks (w) is a straight line and the graph of the sequence formed by the blue blocks (b) is a parabola.
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