A sequence is an ordered set of (usually) numbers .

In the context of sequences, what is a term ?

A term is one of the numbers in a sequence.

In the context of sequences, what is n ?

n is the position of a term in a sequence. E.g. when n = 3, it is referring to the third term of the sequence.

True or False? For the first term , n = 0.

False. For the first term, n = 1.

What is a position-to-term rule?

A position-to-term rule gives the n th term of a sequence as a formula in terms of n .

What is a term-to-term rule?

A term-to-term rule tells you how to find a term from the term before it. I.e., it gives the ( n +1)th term in terms of the n th term.

What is an exponential sequence ?

An exponential sequence is a sequence where one term is multiplied by a common ratio to find the next term . E.g. the sequence: 3, 6, 12, 24, 48, ... is an exponential sequence with the common ratio 2. An exponential sequence is also known as a geometric sequence .

How can a cubic sequence be identified from common differences ?

A cubic sequence can be identified by having a constant third difference . E.g. Consider the sequence: 1, 7, 17, 33, 57, ... The first differences are: 6, 10, 16, 24 The second differences are: 4, 6, 8 The third differences are: 2, 2 The third differences are constant, therefore it is a cubic sequence.

What type of sequence is: 1, 3, 6, 10, 15, ... an example of?

The sequence: 1, 3, 6, 10, 15, is the sequence of triangular numbers . (This is also an example of a quadratic sequence.)

True or False? If the common ratio in an exponential sequence is less than 1 but greater than 0 , then it is an increasing sequence.

False. If the common ratio in an exponential sequence is less than 1 but greater than 0 , then it is a decreasing sequence. It is an increasing sequence if the common ratio is greater than 1 .

What is a linear sequence ?

A linear sequence is a sequence of numbers that increase or decrease by the same amount from one term to the next. A linear sequence is often called an arithmetic sequence .

Define the common difference of a linear sequence.

The common difference is the amount that a linear sequence increases or decreases by from one term to the next .

What is a quadratic sequence ?

A quadratic sequence has an n th term formula that involves n 2 .

True or False? The first differences in a quadratic sequence are constant .

False. The first differences in a quadratic sequence are not constant . The first differences form a linear sequence , which means that the second differences of a quadratic sequence are constant .

How do you find the n th term rule for a quadratic sequence ? E.g. 3, 6, 11, 18, 27, ...

To find the n th term rule for a quadratic sequence , compare the original sequence to the sequence of square numbers . E.g. each value in the sequence, 3, 6, 11, 18, 27, ... is 2 more than the sequence of square numbers, 1, 4, 9, 16, 25, ... So if the sequence of square numbers is n 2 , the original sequence is n 2 + 2.

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FrontIntroduction to Sequences
What is a sequence?

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What is a sequence?
A sequence is an ordered set of (usually) numbers.
In the context of sequences, what is a term?
A term is one of the numbers in a sequence.
In the context of sequences, what is n?
n is the position of a term in a sequence.
E.g. when n = 3, it is referring to the third term of the sequence.
True or False?
For the first term, n = 0.
False.
For the first term, n = 1.
What is subscript notation for sequences?
Subscript notation is used to talk about a particular term.
For example
- $a_{1}$ is the 1st term
- $a_{7}$ is the 7th term
- $a_{n}$ is the nth term
What is a position-to-term rule?
A position-to-term rule gives the nth term of a sequence as a formula in terms of n.
How would you find the first three terms of a sequence using a position-to-term rule?
To find the first three terms of a sequence using a position-to-term rule, substitute $n = 1$ , $n = 2$ and $n = 3$ into the position-to-term formula.
What is a term-to-term rule?
A term-to-term rule tells you how to find a term from the term before it.
I.e., it gives the (n+1)th term in terms of the nth term.
What is an exponential sequence?
An exponential sequence is a sequence where one term is multiplied by a common ratio to find the next term.
E.g. the sequence: 3, 6, 12, 24, 48, ... is an exponential sequence with the common ratio 2.
An exponential sequence is also known as a geometric sequence.
What is the position-to-term rule for an exponential sequence?
The position-to-term rule (nth term rule) for an exponential sequence is $a \times r^{n - 1}$
Where:
- $a$ is the first term
- $r$ is the common ratio
- $n$ is the position of the term in the sequence
How can a cubic sequence be identified from common differences?
A cubic sequence can be identified by having a constant third difference.
E.g. Consider the sequence: 1, 7, 17, 33, 57, ...
The first differences are: 6, 10, 16, 24
The second differences are: 4, 6, 8
The third differences are: 2, 2
The third differences are constant, therefore it is a cubic sequence.
What type of sequence is: 1, 3, 6, 10, 15, ... an example of?
The sequence: 1, 3, 6, 10, 15, is the sequence of triangular numbers.
(This is also an example of a quadratic sequence.)
True or False?
If the common ratio in an exponential sequence is less than 1 but greater than 0, then it is an increasing sequence.
False.
If the common ratio in an exponential sequence is less than 1 but greater than 0, then it is a decreasing sequence.
It is an increasing sequence if the common ratio is greater than 1.
What is a linear sequence?
A linear sequence is a sequence of numbers that increase or decrease by the same amount from one term to the next.
A linear sequence is often called an arithmetic sequence.
Define the common difference of a linear sequence.
The common difference is the amount that a linear sequence increases or decreases by from one term to the next.
What is $d$ the notation for in the context of linear sequences?
$d$ is the notation for the common difference of a linear sequence.
E.g. for a sequence 3, 7, 11, 15, 19, ...
$d = 4$ .
What is $b$ the notation for in the context of linear sequences?
$b$ is the value before the first term (sometimes known as the zero term).
E.g. for a sequence 3, 7, 11, 15, 19, ...
The common difference is +4, so imagine going back from the first term by subtracting 4. So $b = - 1$ .
What is the position-to-term formula for a linear sequence in terms of $b$ , $d$ and $n$ ?
The position-to-term formula (also known as the nth term rule) for a linear sequence in terms of $b$ , $d$ and $n$ is: $n th term = d n + b$ .
What is a quadratic sequence?
A quadratic sequence has an n th term formula that involves n² .
True or False?
The first differences in a quadratic sequence are constant.
False.
The first differences in a quadratic sequence are not constant.
The first differences form a linear sequence, which means that the second differences of a quadratic sequence are constant.
How do you find the n^th term rule for a quadratic sequence?
E.g. 3, 6, 11, 18, 27, ...
To find the nth term rule for a quadratic sequence, compare the original sequence to the sequence of square numbers.
E.g. each value in the sequence, 3, 6, 11, 18, 27, ... is 2 more than the sequence of square numbers, 1, 4, 9, 16, 25, ...
So if the sequence of square numbers is n², the original sequence is n² + 2.
True or False?
For a simple quadratic sequence of the form $a n^{2} + b$ , the value of $a$ is twice the value of the second difference.
False.
For a simple quadratic sequence of the form $a n^{2} + b$ , the value of $a$ is half the value of the second difference.
E.g. for the quadratic sequence: 5, 11, 21, 35, 53, ...
The first differences are: 6, 10, 14, 18
The second differences are: 4, 4, 4
Therefore, $a = 2$ .
To find $b$ , compare the original sequence to the sequence given by $2 n^{2}$ .
The nth term rule for the original sequence is $2 n^{2} + 3$ .

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