Sequences
Sequences are an important area of mathematics, and understanding how to work with them is essential for your Maths exam. In this guide, we will explore various types of sequences, including term-to-term rules, position-to-term rules, arithmetic sequences, quadratic sequences, geometric sequences, and special sequences. You’ll also learn how to find and use the nth term of a sequence.
Term-to-Term Rules
A term-to-term rule describes how to move from one term to the next in a sequence. It tells you what to do to get from one term to the next term, such as adding, subtracting, multiplying, or dividing.
Example 1: Term-to-Term Rule
Consider the sequence:
3, 6, 9, 12, 15, ...
- Term-to-term rule: Add 3 to each term to get the next term.
- In this case, the term-to-term rule is: Add 3 to each term.
Position-to-Term Rules
A position-to-term rule expresses the term in the sequence as a formula depending on its position (n). For example, the nth term formula for a sequence can be used to find any term in the sequence without having to list all the terms.
Example 2: Position-to-Term Rule
Consider the sequence:
2, 4, 6, 8, 10, ...
- The position-to-term rule can be written as:
$$\text{Term} = 2n$$
Where $n$ is the position in the sequence.
- For $n=1$, the term is $2 \times 1 = 2$.
- For $n = 2$, the term is $2 \times 2 = 4$, and so on.
Working Out Position-to-Term Rules for Arithmetic Sequences
An arithmetic sequence is a sequence where each term after the first is found by adding a constant (called the common difference) to the previous term.
Example 3: Finding the Position-to-Term Rule for an Arithmetic Sequence
Consider the sequence:
5, 8, 11, 14, 17, ...
The common difference is $+3$, because each term increases by 3.
To find the position-to-term rule, use the formula:
$$\text{Term} = \text{first term} + (n - 1) \times \text{common difference}$$
In this case, the first term is 5, and the common difference is 3, so the formula is:
$$\text{Term} = 5 + (n - 1) \times 3$$
Simplifying:
$$\text{Term} = 5 + 3n – 3$$
$$\text{Term} = 3n + 2$$
Therefore, the position-to-term rule is $$\text{Term} = 3n + 2$$.
Checking:
- For $n = 1$, the term is $3(1) + 2 = 5$, which is correct.
- For $n = 2$, the term is $3(2) + 2 = 8$, which is correct, and so on.
This video explains number sequences in more detail.
The nth Term
The nth term of a sequence is a formula that allows you to find the value of any term in the sequence, where nnn is the position of the term.
For an arithmetic sequence, the nth term is given by:
$$\text{nth term} = a + (n - 1) \times d$$
Where:
- $a$ is the first term of the sequence.
- $d$ is the common difference.
- $n$ is the position of the term.
Example 4: Finding the nth Term of an Arithmetic Sequence
Consider the sequence:
7, 11, 15, 19, 23, ...
- The first term is $a = 7$.
- The common difference is $d = 4$.
Using the formula for the nth term:
$$\text{nth term} = 7 + (n - 1) \times 4$$
Simplifying:
$$\text{nth term} = 7 + 4n – 4$$
$$\text{nth term} = 4n + 3$$
So, the nth term is $\text{Term} = 4n + 3$
Using the nth Term
Once you have the nth term formula, you can use it to find any term in the sequence. Simply substitute the position nnn into the formula.
Example 5: Using the nth Term Formula
Using the nth term formula $\text{Term} = 4n + 3$, find the 6th term of the sequence.
Substitute $n = 6$ into the formula:
$$\text{Term} = 4(6) + 3$$
$$\text{Term} = 24 + 3 = 27$$
Therefore, the 6th term is 27.
Finding the nth Term of Quadratic Sequences
A quadratic sequence is a sequence where the difference between consecutive terms changes in a regular pattern. The second difference (the difference between the differences) is constant in a quadratic sequence.
To find the nth term of a quadratic sequence, follow these steps:
- Find the first and second differences.
- Use the second difference to determine the coefficient of $n^2$.
- Find the other coefficients by using the first few terms.
Example 6: Finding the nth Term of a Quadratic Sequence
Consider the sequence:
1, 4, 9, 16, 25, ...
The first differences are:
$4−1=3, 9−4=5, 16−9=7, 25−16=9$.
The second differences are constant:
$5−3=2, 7−5=2, 9−7=2$.
Since the second difference is 2, the coefficient of $n^2$ is half of this value, i.e. $1$.
Now, the nth term formula starts as:
$$\text{nth term} = n^2 + \text{linear term} + \text{constant}$$
Substitute the first few terms into the formula to find the linear and constant terms. After working through the calculations, you find that the nth term is:
$$\text{nth term} = n^2$$
So, the nth term for this sequence is $n^2$.
Geometric Sequences
A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant (called the common ratio).
The general form of the nth term for a geometric sequence is:
$$\text{nth term} = a \times r^{(n-1)}$$
Where:
- $a$ is the first term.
- $r$ is the common ratio.
- $n$ is the position of the term.
Example 7: Geometric Sequence
Consider the geometric sequence:
2, 6, 18, 54, 162, ...
- The first term is $a = 2$.
- The common ratio is $r = 3$.
Using the formula for the nth term:
$$\text{nth term} = 2 \times 3^{(n-1)}$$
So, the nth term is $\text{Term} = 2 \times 3^{(n-1)}$.
Special Sequences
Some sequences don’t follow the rules of arithmetic, geometric, or quadratic sequences but are still important. Special sequences include:
- Fibonacci Sequence: Each term is the sum of the two previous terms.
- Square Numbers: $1,4,9,16,25,….$
- Cube Numbers: $1,8,27,64,125,….$
Example 8: Fibonacci Sequence
The Fibonacci sequence starts as:
1, 1, 2, 3, 5, 8, 13, 21, ...
Each term is the sum of the two previous terms. There isn’t a simple nth term formula for the Fibonacci sequence, but it’s important to recognise the pattern.
Summary of Key Points
- Term-to-term rule: Describes how to get from one term to the next.
- Position-to-term rule: A formula for finding any term in the sequence based on its position.
- Arithmetic sequence: A sequence with a constant difference between terms. nth term formula: $\text{Term} = a + (n - 1) \times d$.
- Quadratic sequence: A sequence where the second difference is constant. nth term formula involves $n^2$.
- Geometric sequence: A sequence where each term is multiplied by a constant ratio. nth term formula: $\text{Term} = a \times r^{(n-1)}$.
- Special sequences: Includes Fibonacci, square numbers, and cube numbers.
By understanding and practising these concepts, you'll be well-prepared to tackle any sequence problem in your GCSE Maths exam.
