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.

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 (Cambridge (CIE) IGCSE International Maths: Core): Flashcards

Exam code: 0607

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FrontIntroduction to Sequences
What is a sequence?

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Cards in this collection (27)

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.
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: $n th term = d n + b$ .
Define a quadratic sequence.
A quadratic sequence is one whose $n$ th term formula contains $n^{2}$ .
For example, $2, 5, 10, 17, 26, \dots$ has $n$ th term $n^{2} + 1$ .
Complete the test for each kind of sequence, using the correct ordinal in each gap.
A sequence is quadratic when its $______$ differences are constant, and cubic when its $______$ differences are constant instead.
The completed test is:
A sequence is quadratic when its second differences are constant, and cubic when its third differences are constant instead.
The second differences are the differences of the first differences, and the third differences are the differences of those.
Once you know a sequence is quadratic, how do you find its $n$ th term formula?
Write the square numbers $1, 4, 9, 16, 25, \dots$ underneath it and compare the two term by term.
If every term is the same amount more, add that amount to $n^{2}$ ; if every term is a multiple, multiply $n^{2}$ by it.
What is the $n$ th term of the sequence $6, 9, 14, 21, 30$ and so on?
It is $n^{2} + 5$ .
The second differences are all $2$ , and every term is $5$ more than the matching square number.
True or False?
The sequence $16, 25, 36, 49, \dots$ has $n$ th term $n^{2}$ .
False.
These are the square numbers starting from $4^{2}$ , so the formula is ${(n + 3)}^{2}$ .
Substituting $n = 1$ gives $4^{2} = 16$ , which is indeed the first term.
In a sequence with $n$ th term $a n^{2} + b$ , how do you find $a$ from the differences?
Take half of the second difference.
For the cubic form $a n^{3} + b$ you take a sixth of the third difference instead.
Which numbers do you compare a cubic sequence with to find its formula?
You compare it with the cube numbers $1, 8, 27, 64, 125, \dots$ , which are $1^{3}, 2^{3}, 3^{3}, 4^{3}, \dots$
For instance $2, 9, 28, 65, 126, \dots$ is each cube number plus $1$ , so its $n$ th term is $n^{3} + 1$ .
True or False?
Two different quadratic sequences can have exactly the same second differences.
True.
The second difference fixes only the number in front of $n^{2}$ , so $n^{2} + 1$ and $n^{2} + 5$ both have second differences of $2$ .
That is why the sequence still has to be compared with the square numbers.

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