Arithmetic and Geometric Progressions

Identifying Types of Sequences

You can save money by adding £5 to a jar every week, or you could watch bacteria multiply by doubling every hour. These two types of growth represent the two fundamental types of mathematical sequences.
To identify a sequence type, you must test the relationship between consecutive terms using a term-to-term rule, which describes how to get from one term directly to the next.
An arithmetic progression (often called a linear sequence by AQA) has a constant difference between consecutive terms.
A geometric progression (also known as an exponential sequence) has a constant multiplier between consecutive terms.
Testing Method:
1. Calculate $U_2 - U_1$ and $U_3 - U_2$ . If they are the same, it is an arithmetic progression.
2. Calculate $U_2 \div U_1$ and $U_3 \div U_2$ . If they are the same, it is a geometric progression.

Arithmetic Progressions

The fixed value added to or subtracted from each term is called the common difference ( $d$ ).
The position of a term in the sequence is denoted by $n$ , which must always be a positive integer ( $n=1$ for the 1st term, $n=2$ for the 2nd term, etc.).
The relationship between the position ( $n$ ) and the value of the term is given by the position-to-term rule or the $n^{th}$ term formula.
For GCSE, the standard formula is $dn + b$ , where $d$ is the common difference and $b$ is the "zero term" (calculated as the first term minus $d$ ).

Worked Example: Arithmetic $n^{th}$ Term

Identify the sequence type for $5, 7, 9, 11, \dots$ , find its $n^{th}$ term, and calculate the 50th term.

Step 1: Test the sequence to identify its type and find the common difference ( $d$ ).

$7 - 5 = 2$ and $9 - 7 = 2$
The differences are equal, so it is an arithmetic progression with $d = 2$ .
The formula starts with $2n$ .

Step 2: Find the zero term ( $b$ ).

$b = \text{First term} - d$
$b = 5 - 2 = 3$

Step 3: Write the full $n^{th}$ term formula.

$n^{th} \text{ term} = 2n + 3$

Step 4: Calculate the 50th term by substituting $n = 50$ .

$50^{th} \text{ term} = 2(50) + 3 = 103$

Geometric Progressions

In a geometric progression, each term is found by multiplying the previous term by a constant called the common ratio ( $r$ ).
If $r > 1$ , the sequence is increasing. If $0 < r < 1$ , the sequence is decreasing (decay). If $r < 0$ , the terms alternate between positive and negative.
For Higher Tier, you must use the formal $n^{th}$ term formula:

u_n = ar^{n-1}

$a$ represents the first term of the sequence.
The power is $(n - 1)$ because the first term (where $n = 1$ ) has $r^0 = 1$ .

Worked Example: Geometric $n^{th}$ Term (Higher Tier)

Identify the sequence type for $4, 12, 36, 108, \dots$ , find its $n^{th}$ term, and calculate the 6th term.

Step 1: Test the sequence to identify its type.

$12 \div 4 = 3$ and $36 \div 12 = 3$
The ratios are equal, so it is a geometric progression.

Step 2: Identify the first term ( $a$ ) and common ratio ( $r$ ).

$a = 4$
$r = 3$

Step 3: Substitute into the general formula $u_n = ar^{n-1}$ .

$n^{th} \text{ term} = 4 \times 3^{n-1}$

Step 4: Calculate the 6th term by substituting $n = 6$ .

$u_6 = 4 \times 3^{6-1} = 4 \times 3^5$
$u_6 = 4 \times 243 = 972$

Verifying a Term in a Sequence

Exam questions often ask if a specific value exists within a given arithmetic sequence.
To verify this, set your $n^{th}$ term formula equal to the value and solve for $n$ .
If $n$ is a positive integer, the value is in the sequence. If $n$ is a fraction or decimal, the value is NOT in the sequence.

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Exam Tips

Common Mistake
Students often multiply 'a' and 'r' together before applying the power in geometric sequences (e.g., writing $4 \times 3^{n-1}$ as $12^{n-1}$ ). Due to BIDMAS/BODMAS, you must calculate the index before multiplying!
2
Always check your arithmetic $n^{th}$ term formula by substituting $n=1$ and $n=2$ to ensure they produce the actual first two terms of the given sequence.
3
AQA frequently presents arithmetic sequences as visual patterns of tiles or matchsticks; always count the items and write out the numerical sequence first before calculating the $n^{th}$ term.
4
For geometric trial and error questions on the calculator paper, use the 'ANS' key: type the first term, press '=', then type 'x [r]' and press '=' repeatedly to generate terms rapidly.

Key Terms(9)

Term-to-term rule: A rule describing the mathematical operation needed to get from one term directly to the next in a sequence.
Arithmetic progression: A sequence of numbers where the difference between any two consecutive terms is constant.
Linear sequence: The exact same mathematical concept as an arithmetic progression, characterised by a constant difference between terms.
Geometric progression: A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number.
Exponential sequence: Another name for a geometric progression, reflecting the fact that its growth is based on repeated multiplication.
Common difference: The constant amount added to each term in an arithmetic progression to get the next term, denoted by 'd'.
Position-to-term rule: A formula (often called the nth term) that defines the relationship between a term's position (n) and its actual value.
Common ratio: The constant multiplier between consecutive terms in a geometric progression, denoted by 'r'.
First term: The starting value of a sequence, often represented by 'a' in Higher Tier geometric formulas.

Previous NoteCommon Number Patterns and Fibonacci Sequences Next NoteQuadratic and Surd Sequences

Back to Special Sequences

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(9)

Term-to-term rule: A rule describing the mathematical operation needed to get from one term directly to the next in a sequence.
Arithmetic progression: A sequence of numbers where the difference between any two consecutive terms is constant.
Linear sequence: The exact same mathematical concept as an arithmetic progression, characterised by a constant difference between terms.
Geometric progression: A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number.
Exponential sequence: Another name for a geometric progression, reflecting the fact that its growth is based on repeated multiplication.
Common difference: The constant amount added to each term in an arithmetic progression to get the next term, denoted by 'd'.
Position-to-term rule: A formula (often called the nth term) that defines the relationship between a term's position (n) and its actual value.
Common ratio: The constant multiplier between consecutive terms in a geometric progression, denoted by 'r'.
First term: The starting value of a sequence, often represented by 'a' in Higher Tier geometric formulas.

Arithmetic and Geometric Progressions

Identifying Types of Sequences

You can save money by adding £5 to a jar every week, or you could watch bacteria multiply by doubling every hour. These two types of growth represent the two fundamental types of mathematical sequences.
To identify a sequence type, you must test the relationship between consecutive terms using a term-to-term rule, which describes how to get from one term directly to the next.
An arithmetic progression (often called a linear sequence by AQA) has a constant difference between consecutive terms.
A geometric progression (also known as an exponential sequence) has a constant multiplier between consecutive terms.
Testing Method:
1. Calculate $U_2 - U_1$ and $U_3 - U_2$ . If they are the same, it is an arithmetic progression.
2. Calculate $U_2 \div U_1$ and $U_3 \div U_2$ . If they are the same, it is a geometric progression.

Arithmetic Progressions

The fixed value added to or subtracted from each term is called the common difference ( $d$ ).
The position of a term in the sequence is denoted by $n$ , which must always be a positive integer ( $n=1$ for the 1st term, $n=2$ for the 2nd term, etc.).
The relationship between the position ( $n$ ) and the value of the term is given by the position-to-term rule or the $n^{th}$ term formula.
For GCSE, the standard formula is $dn + b$ , where $d$ is the common difference and $b$ is the "zero term" (calculated as the first term minus $d$ ).

Worked Example: Arithmetic $n^{th}$ Term

Identify the sequence type for $5, 7, 9, 11, \dots$ , find its $n^{th}$ term, and calculate the 50th term.

Step 1: Test the sequence to identify its type and find the common difference ( $d$ ).

$7 - 5 = 2$ and $9 - 7 = 2$
The differences are equal, so it is an arithmetic progression with $d = 2$ .
The formula starts with $2n$ .

Step 2: Find the zero term ( $b$ ).

$b = \text{First term} - d$
$b = 5 - 2 = 3$

Step 3: Write the full $n^{th}$ term formula.

$n^{th} \text{ term} = 2n + 3$

Step 4: Calculate the 50th term by substituting $n = 50$ .

$50^{th} \text{ term} = 2(50) + 3 = 103$

Geometric Progressions

In a geometric progression, each term is found by multiplying the previous term by a constant called the common ratio ( $r$ ).
If $r > 1$ , the sequence is increasing. If $0 < r < 1$ , the sequence is decreasing (decay). If $r < 0$ , the terms alternate between positive and negative.
For Higher Tier, you must use the formal $n^{th}$ term formula:

u_n = ar^{n-1}

$a$ represents the first term of the sequence.
The power is $(n - 1)$ because the first term (where $n = 1$ ) has $r^0 = 1$ .

Worked Example: Geometric $n^{th}$ Term (Higher Tier)

Identify the sequence type for $4, 12, 36, 108, \dots$ , find its $n^{th}$ term, and calculate the 6th term.

Step 1: Test the sequence to identify its type.

$12 \div 4 = 3$ and $36 \div 12 = 3$
The ratios are equal, so it is a geometric progression.

Step 2: Identify the first term ( $a$ ) and common ratio ( $r$ ).

$a = 4$
$r = 3$

Step 3: Substitute into the general formula $u_n = ar^{n-1}$ .

$n^{th} \text{ term} = 4 \times 3^{n-1}$

Step 4: Calculate the 6th term by substituting $n = 6$ .

$u_6 = 4 \times 3^{6-1} = 4 \times 3^5$
$u_6 = 4 \times 243 = 972$

Verifying a Term in a Sequence

Exam questions often ask if a specific value exists within a given arithmetic sequence.
To verify this, set your $n^{th}$ term formula equal to the value and solve for $n$ .
If $n$ is a positive integer, the value is in the sequence. If $n$ is a fraction or decimal, the value is NOT in the sequence.

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often multiply 'a' and 'r' together before applying the power in geometric sequences (e.g., writing $4 \times 3^{n-1}$ as $12^{n-1}$ ). Due to BIDMAS/BODMAS, you must calculate the index before multiplying!
2
Always check your arithmetic $n^{th}$ term formula by substituting $n=1$ and $n=2$ to ensure they produce the actual first two terms of the given sequence.
3
AQA frequently presents arithmetic sequences as visual patterns of tiles or matchsticks; always count the items and write out the numerical sequence first before calculating the $n^{th}$ term.
4
For geometric trial and error questions on the calculator paper, use the 'ANS' key: type the first term, press '=', then type 'x [r]' and press '=' repeatedly to generate terms rapidly.

Key Terms(9)

Term-to-term rule: A rule describing the mathematical operation needed to get from one term directly to the next in a sequence.
Arithmetic progression: A sequence of numbers where the difference between any two consecutive terms is constant.
Linear sequence: The exact same mathematical concept as an arithmetic progression, characterised by a constant difference between terms.
Geometric progression: A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number.
Exponential sequence: Another name for a geometric progression, reflecting the fact that its growth is based on repeated multiplication.
Common difference: The constant amount added to each term in an arithmetic progression to get the next term, denoted by 'd'.
Position-to-term rule: A formula (often called the nth term) that defines the relationship between a term's position (n) and its actual value.
Common ratio: The constant multiplier between consecutive terms in a geometric progression, denoted by 'r'.
First term: The starting value of a sequence, often represented by 'a' in Higher Tier geometric formulas.

Previous NoteCommon Number Patterns and Fibonacci Sequences Next NoteQuadratic and Surd Sequences

Back to Special Sequences

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(9)

Term-to-term rule: A rule describing the mathematical operation needed to get from one term directly to the next in a sequence.
Arithmetic progression: A sequence of numbers where the difference between any two consecutive terms is constant.
Linear sequence: The exact same mathematical concept as an arithmetic progression, characterised by a constant difference between terms.
Geometric progression: A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number.
Exponential sequence: Another name for a geometric progression, reflecting the fact that its growth is based on repeated multiplication.
Common difference: The constant amount added to each term in an arithmetic progression to get the next term, denoted by 'd'.
Position-to-term rule: A formula (often called the nth term) that defines the relationship between a term's position (n) and its actual value.
Common ratio: The constant multiplier between consecutive terms in a geometric progression, denoted by 'r'.
First term: The starting value of a sequence, often represented by 'a' in Higher Tier geometric formulas.

Arithmetic and Geometric Progressions

Identifying Types of Sequences

Arithmetic Progressions

Worked Example: Arithmetic nthn^{th}nth Term

Geometric Progressions

Worked Example: Geometric nthn^{th}nth Term (Higher Tier)

Verifying a Term in a Sequence

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Exam Tips

Key Terms(9)

Key Terms(9)

Arithmetic and Geometric Progressions

Identifying Types of Sequences

Arithmetic Progressions

Worked Example: Arithmetic nthn^{th}nth Term

Geometric Progressions

Worked Example: Geometric nthn^{th}nth Term (Higher Tier)

Verifying a Term in a Sequence

Sign up to continue reading

Exam Tips

Key Terms(9)

Key Terms(9)

Worked Example: Arithmetic $n^{th}$ Term

Worked Example: Geometric $n^{th}$ Term (Higher Tier)

Worked Example: Arithmetic $n^{th}$ Term

Worked Example: Geometric $n^{th}$ Term (Higher Tier)