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 the first thing to work out when you are asked to continue a sequence?

Work out the first differences , the amounts the sequence changes by from one term to the next. Writing them underneath the sequence makes any pattern much easier to see.

True or False? If the first differences of a sequence are not all the same, the sequence cannot be continued.

False. When the first differences are not all the same, look for a pattern in the differences themselves and carry that pattern on. A sequence whose differences change can still be continued perfectly well.

How can you tell a quadratic sequence, a geometric sequence and a Fibonacci sequence apart?

Look at how each term is produced from the ones before it. A quadratic sequence has constant second differences, a geometric sequence multiplies by a constant each time, and a Fibonacci sequence adds the previous two terms.

Why do you need two starting terms before a Fibonacci sequence can be generated?

Because each term is made by adding the two terms before it. With only one starting value there is nothing to add it to, so no further term can be worked out.

True or False? You find the common ratio of a geometric sequence by subtracting one term from the next.

False. You divide a term by the one before it, and any consecutive pair will do. Subtracting gives a common difference, which belongs to a linear sequence rather than a geometric one.

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 .

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Sequences (Edexcel GCSE Maths: Foundation): Flashcards

Exam code: 1MA1

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FrontIntroduction to Sequences
What is a sequence?

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Cards in this collection (25)

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 the first thing to work out when you are asked to continue a sequence?
Work out the first differences, the amounts the sequence changes by from one term to the next.
Writing them underneath the sequence makes any pattern much easier to see.
True or False?
If the first differences of a sequence are not all the same, the sequence cannot be continued.
False.
When the first differences are not all the same, look for a pattern in the differences themselves and carry that pattern on.
A sequence whose differences change can still be continued perfectly well.
What is the next term of $6, 1, - 4, - 9$ ?
The next term is $- 14$ .
The first differences are all $- 5$ , so subtracting another $5$ from $- 9$ gives $- 14$ .
In $2, 8, 15, 23$ the first differences are $+ 6, + 7, + 8$ . What is the next term?
The next term is $32$ .
The differences go up by $1$ each time, so the next difference is $+ 9$ , and $23 + 9 = 32$ .
In $1, 3, 7, 15$ the first differences are $+ 2, + 4, + 8$ . Complete the next step:
the next difference is $+______$ and the next term is $______$
The completed step is:
the next difference is $+ 16$ and the next term is $31$
Each difference is double the one before it, and $15 + 16 = 31$ .
Define a triangular number.
A triangular number is the result of adding consecutive whole numbers starting from $1$ , giving $1$ , $1 + 2$ , $1 + 2 + 3$ , and so on.
The first five are $1, 3, 6, 10, 15$ , and they are so called because that many dots can be arranged into a triangle.
How can you tell a quadratic sequence, a geometric sequence and a Fibonacci sequence apart?
Look at how each term is produced from the ones before it.
A quadratic sequence has constant second differences, a geometric sequence multiplies by a constant each time, and a Fibonacci sequence adds the previous two terms.
The first differences of $6, 10, 16, 24, 34$ are $+ 4, + 6, + 8, + 10$ . Complete the working:
the second differences are all $+______$ and the next term is $______$
The completed working is:
the second differences are all $+ 2$ and the next term is $46$
Constant second differences are the sign of a quadratic sequence, and the next first difference is $+ 12$ .
What is the common ratio of the geometric sequence $4, 8, 16, 32$ ?
The common ratio is $2$ .
Each term is twice the one before it, so the sequence continues $64$ , $128$ , and so on.
True or False?
A sequence can be called a Fibonacci sequence even if it does not begin $1, 1$ .
True.
Any sequence in which each term is the sum of the previous two is a Fibonacci sequence, such as $2, 9, 11, 20, 31$ .
The Fibonacci sequence is the particular one that begins $1, 1, 2, 3, 5, 8$ .
Why do you need two starting terms before a Fibonacci sequence can be generated?
Because each term is made by adding the two terms before it.
With only one starting value there is nothing to add it to, so no further term can be worked out.
True or False?
You find the common ratio of a geometric sequence by subtracting one term from the next.
False.
You divide a term by the one before it, and any consecutive pair will do.
Subtracting gives a common difference, which belongs to a linear sequence rather than a geometric one.
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: $n th term = d n + b$ .

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