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.

True or False? The term-to-term rule for a linear sequence is to add the same amount each time.

True. The term-to-term rule for a linear sequence is to add the same amount each time.

True or False? The term-to-term rule for a geometric sequence is to multiply by the term before each time.

False. The term-to-term rule for a geometric sequence is to multiply by the same amount each time (called a common ratio), not by the term before.

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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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.
True or False?
The term-to-term rule for a linear sequence is to add the same amount each time.
True.
The term-to-term rule for a linear sequence is to add the same amount each time.
True or False?
The term-to-term rule for a geometric sequence is to multiply by the term before each time.
False.
The term-to-term rule for a geometric sequence is to multiply by the same amount each time (called a common ratio), not by the term before.
True or False?
$- 8, 8, 0, 8, 16, . . .$ is a Fibonacci sequence.
False.
$- 8, 8, 0, 8, 16, . . .$ is not a Fibonacci sequence.
The sum of the two terms before the 16 is 0 + 8, which equals 8 (not 16).
Explain how to find the common ratio of the geometric sequence 4, 14, 49, ...
The common ratio of a geometric sequence is the amount you multiply by each time.
Find out what you multiply 4 by to get 14. This is $\frac{14}{4} = 3.5$ .
Check that this works for the next two terms: $14 \times 3.5 = 49$ (correct).
The common ratio is 3.5.
True or False?
$\sqrt{3}, 3, \sqrt{27}, 9, . . .$ is a geometric sequence.
True.
$\sqrt{3}, 3, \sqrt{27}, 9, . . .$ is a geometric sequence.
It helps to use surds to simplify the 3^rd term: $\sqrt{27} = \sqrt{9} \times \sqrt{3} = 3 \sqrt{3}$ .
Now the sequence is $\sqrt{3}, 3, 3 \sqrt{3}, 9, . . .$
Remember that $3 = \sqrt{3} \times \sqrt{3}$ so the common ratio is $\sqrt{3}$ .
You can check this by writing it as $\sqrt{3}, {(\sqrt{3})}^{2}, {(\sqrt{3})}^{3}, {(\sqrt{3})}^{4}, . . .$
True or False?
$a, b, a + b, a + 2 b, . . .$ is a Fibonacci sequence.
True.
$a, b, a + b, a + 2 b, . . .$ is a Fibonacci sequence.
Add the first two terms to get $a + b$ (the correct 3^rd term).
Add the 2^nd and 3^rd term to get $b + (a + b)$ which gives $a + 2 b$ (the correct 4^th term).
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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