
Quadratic Sequences
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3.2 Quadratic sequences
Quadratic sequences
- Study the dotted-tile pattern shown and answer the following questions.
- Complete the fourth pattern in the diagram.
-
Complete the table below:
pattern number dotted tiles difference ( )
- What do you notice about the change in number of dotted tiles?
- Describe the pattern in words: “The number of dotted tiles...”.
- Write the general term:
- Give the mathematical name for this kind of pattern.
- A pattern has
dotted tiles. Determine the value of
.
- Now study the number of blank tiles (tiles without dots) and answer the following questions:
-
Complete the table below:
pattern number blank tiles first difference second difference - What do you notice about the change in the number of blank tiles?
- Describe the pattern in words: “The number of blank tiles...”.
- Write the general term:
- Give the mathematical name for this kind of pattern.
- A pattern has
blank tiles. Determine the value of
.
- A pattern has
dotted tiles. Determine the number of blank tiles.
-
- Quadratic sequence
-
A quadratic sequence is a sequence of numbers in which the second difference between any two consecutive terms is constant.
Consider the following example:
The first difference is calculated by finding the difference between consecutive terms:
The second difference is obtained by taking the difference between consecutive first differences:
We notice that the second differences are all equal to . Any sequence that has a common second difference is a quadratic sequence.
It is important to note that the first differences of a quadratic sequence form a sequence. This sequence has a constant difference between consecutive terms. In other words, a linear sequence results from taking the first differences of a quadratic sequence.
General case
If the sequence is quadratic, the term is of the form
.
In each case, the common second difference is a .
Determine the second difference between the terms for the following sequences:











Complete the sequence by filling in the missing term:
Use the general term to generate the first four terms in each sequence:




Worked example 2: Quadratic sequences
Write down the next two terms and determine an equation for the term of the sequence
;
;
;
;
Find the second differences between the terms
So there is a common second difference of . We can therefore conclude that this is a quadratic sequence of the form
.
Continuing the sequence, the next first differences will be:
Finding the next two terms in the sequence
So the sequence will be: ;
;
;
;
;
;
Determine the general term for the sequence
To find the values of ,
and
for
we look at the first
terms in the sequence:

We solve a set of simultaneous equations to determine the values of ,
and
We know that ,
and





Write the general term for the sequence
Worked example 3: Plotting a graph of terms in a sequence
Consider the following sequence:
- Determine the general term (
) for the sequence.
- Is this a linear or a quadratic sequence?
- Plot a graph of
vs
.
Determine the first and second differences
We see that the first differences are not constant and form the sequence and that there is a common second difference of
. Therefore the sequence is quadratic and has a general term of the form
.
Determine the general term 
To find the values of ,
and
for
we look at the first
terms in the sequence:

We solve this set of simultaneous equations to determine the values of ,
and
. We know that
,
and
.





Therefore the general term for the sequence is .
Plot a graph of
vs 
Use the general term for the sequence, , to complete the table.
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Use the table to plot the graph:
In this case it would not be accurate to join these points, since indicates the position of a term in a sequence and can therefore only be a positive integer. We can, however, see that the plot of the points lies in the shape of a parabola.
Worked example 4: Olympic Games soccer event
In the first stage of the soccer event at the Olympic Games, there are teams from four different countries in each group. Each country in a group must play every other country in the group once.
- How many matches will be played in each group in the first stage of the event?
- How many matches would be played if there are
teams in each group?
- How many matches would be played if there are
teams in each group?
- Determine the general formula of the sequence.
Determine the number of matches played if there are
teams in a group
Let the teams from four different countries be ,
,
and
.
teams in a group | matches played |
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means that team
plays team
and
would be the same match as
. So if there are four different teams in a group, each group plays
matches.
Determine the number of matches played if there are
teams in a group
Let the teams from five different countries be and
.
teams in a group | matches played |
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So if there are five different teams in a group, each group plays matches.
Determine the number of matches played if there are
teams in a group
Let the teams from six different countries be and
.
teams in a group | matches to be played |
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So if there are six different teams in a group, each group plays matches.
We continue to increase the number of teams in a group and find that a group of teams plays
matches and a group of
teams plays
matches.
Consider the sequence
We examine the sequence to determine if it is linear or quadratic:
We see that the first differences are not constant and that there is a common second difference of . Therefore the sequence is quadratic and has a general term of the form
.
Determine the general term 
To find the values of ,
and
for
we look at the first
terms in the sequence:

We solve a set of simultaneous equations to determine the values of ,
and
. We know that
,
and





Therefore the general term for the sequence is .
Calculate the common second difference for each of the following quadratic sequences:





Find the first five terms of the quadratic sequence defined by: .


Given , find
.

Given , for which value of
does
?

Write down the next two terms of the quadratic sequence:
Find the general formula for the quadratic sequence above.



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