Arithmetic and Geometric Progressions

Identifying Arithmetic Progressions

If you save £5 every single week, your total savings grow in a perfectly predictable pattern. This is an example of an arithmetic progression, frequently referred to as a linear sequence in Edexcel papers. The defining feature is that the difference between any two consecutive terms is always constant.

This constant value added or subtracted is the common difference ( $d$ ). You can identify if a sequence is arithmetic by checking if the difference between terms is consistent (i.e., $Term_2 - Term_1 = Term_3 - Term_2$ ). Unlike geometric sequences, arithmetic progressions do not multiply terms by a constant factor; they only ever add or subtract.

The General Formula for the nth Term

The nth term is an algebraic rule used to find the value of a term at any positive integer position $n$ . The first term ( $a$ ) is the starting value of the sequence when $n = 1$ . The general formula must be memorised as it is not provided on the Edexcel formula sheet:

u_n = a + (n - 1)d

Worked Example: Algebraic Simplification

Find the nth term of the sequence $7, 12, 17, 22 \dots$

Step 1: Identify the first term ( $a$ ) and common difference ( $d$ ). $a = 7$ , $d = 12 - 7 = 5$

Step 2: Substitute these values into the general formula. $u_n = 7 + (n - 1)5$

Step 3: Expand the brackets. $u_n = 7 + 5n - 5$

Step 4: Collect like terms to simplify. $u_n = 5n + 2$

Worked Example: Checking a Term

Is $99$ in the sequence $15, 21, 27 \dots$ ?

Step 1: Find the nth term rule using the zero term method. The common difference is $d = 6$ . The "zero term" (one step back from $15$ ) is $15 - 6 = 9$ . The sequence rule is $6n + 9$ .

Step 2: Set the rule equal to the target number. $6n + 9 = 99$

Step 3: Solve for $n$ . $6n = 90$ $n = 15$

Step 4: Interpret the result. Since $n$ is an integer, $99$ is exactly the 15th term in the sequence. If $n$ had been a decimal, the number would not be in the sequence.

Identifying Geometric Progressions

A linear sequence adds a constant amount each time, but a geometric progression (GP) multiplies, leading to rapid explosions in size or rapid decay. Each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio ( $r$ ).

You can find $r$ by dividing any term by the previous term ( $r = \frac{u_{n+1}}{u_n}$ ). If $|r| > 1$ , the sequence shows exponential growth, and if $0 < |r| < 1$ , it shows exponential decay. If $r$ is a negative number, the sequence does not just grow or shrink; the signs of the terms will alternate between positive and negative.

Calculating with Geometric Progressions

The Edexcel specification often represents these sequences simply as $r^n$ , but the formal general formula used to find the nth term of a geometric progression is:

u_n = ar^{n-1}

Higher tier students must be able to handle common ratios that are fractions or a surd (an irrational number left exactly inside a root sign). When substituting fractions into the formula, it is essential to use brackets around the fraction before applying the power.

Worked Example: Finding the nth Term Expression

Find the nth term expression for a geometric progression where the second term is $-320$ and the common ratio is $-\frac{2}{5}$ .

Step 1: Find the first term ( $a$ ) using the formula $u_2 = ar$ . $-320 = a \times (-\frac{2}{5})$ $a = -320 \div (-\frac{2}{5})$ $a = 800$

Step 2: State the general nth term expression using $u_n = ar^{n-1}$ . $u_n = 800 \times (-\frac{2}{5})^{n-1}$

Worked Example: Fractional Ratio

Find the 6th term of the sequence $80, 40, 20, 10 \dots$

Step 1: Find the common ratio ( $r$ ) and first term ( $a$ ). $r = \frac{40}{80} = \frac{1}{2}$ $a = 80$

Step 2: Substitute into the formula for $n=6$ . $u_6 = 80 \times (\frac{1}{2})^{6-1}$

Step 3: Calculate the power. $u_6 = 80 \times (\frac{1}{2})^5$ $u_6 = 80 \times \frac{1}{32}$

Step 4: Find the final answer. $u_6 = \frac{80}{32} = 2.5$

Worked Example: Generating Terms with a Surd Ratio

Find the 5th term of the sequence where the nth term rule is $(\sqrt{3})^n$ .

Step 1: Substitute $n=5$ into the expression. $u_5 = (\sqrt{3})^5$

Step 2: Expand the surd multiplication. $u_5 = \sqrt{3} \times \sqrt{3} \times \sqrt{3} \times \sqrt{3} \times \sqrt{3}$

Step 3: Simplify the pairs of identical surds ( $\sqrt{3} \times \sqrt{3} = 3$ ). $u_5 = 3 \times 3 \times \sqrt{3}$

Step 4: Calculate the final exact form. $u_5 = 9\sqrt{3}$

Worked Example: Finding a Surd Common Ratio (Higher Tier)

Find the common ratio ( $r$ ) of a geometric progression where the first term is $3 + 2\sqrt{2}$ and the second term is $13 + 9\sqrt{2}$ .

Step 1: Use the ratio formula $r = \frac{u_2}{u_1}$ . $r = \frac{13 + 9\sqrt{2}}{3 + 2\sqrt{2}}$

Step 2: Rationalise the denominator by multiplying the numerator and denominator by the conjugate $(3 - 2\sqrt{2})$ . $r = \frac{(13 + 9\sqrt{2})(3 - 2\sqrt{2})}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})}$

Step 3: Expand the brackets for the numerator and denominator. Numerator: $39 - 26\sqrt{2} + 27\sqrt{2} - 36 = 3 + \sqrt{2}$ Denominator: $9 - 6\sqrt{2} + 6\sqrt{2} - 8 = 1$

Step 4: Simplify to find the exact form of $r$ . $r = 3 + \sqrt{2}$

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 use $n=0$ to try and find the first term of a sequence, but the position variable $n$ must always be a positive integer starting at $n=1$ .
2
In 'deduce the nth term' questions, examiners usually award one mark for correctly identifying the common difference (e.g., writing $4n$ ) and a second mark for finding the correct constant.
3
When a linear sequence is decreasing (e.g., $20, 15, 10 \dots$ ), ensure you include the negative sign in your nth term expression (write $-5n$ , not $5n$ ).
4
For geometric progressions, remember the power in the general formula is $n-1$ , not $n$ . Always verify your formula works by checking it against the first term.
5
If calculating a geometric sequence where the common ratio is a fraction or surd, Edexcel mark schemes frequently require the final answer to be left in exact form rather than a rounded decimal.
6
When finding the common ratio of a geometric progression containing surds (Higher Tier), you will often need to rationalise the denominator by multiplying the numerator and denominator by the conjugate.

Key Terms(12)

Arithmetic progression: A sequence where the difference between any two consecutive terms is always constant.
Linear sequence: An alternative name for an arithmetic progression commonly used in Edexcel exam papers.
Common difference: The constant value added to, or subtracted from, each successive term in an arithmetic progression.
nth term: A general algebraic rule or formula used to find the value of a sequence's term at any specific position.
First term: The starting value of a sequence, represented algebraically as 'a' or 'u₁'.
Geometric progression (GP): A sequence where each term is found by multiplying the previous term by a constant, non-zero number.
Common ratio: The constant multiplier between successive terms in a geometric progression.
Surd: An irrational number expressed exactly using a root sign to avoid rounding decimals.
Exponential growth: A process where a quantity increases over time at a rate proportional to its current value, represented by a geometric sequence with a common ratio where |r| > 1.
Exponential decay: A process where a quantity decreases over time at a rate proportional to its current value, represented by a geometric sequence with a common ratio where 0 < |r| < 1.
Rationalise: The process of removing a surd or irrational number from the denominator of a fraction.
Conjugate: A binomial formed by negating the second term of a binomial, often used to rationalise denominators containing surds.

Previous Subtopic: Term GenerationTerm Generation Next NoteQuadratic Sequences

Back to Sequence Types

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

Key Terms(12)

Arithmetic progression: A sequence where the difference between any two consecutive terms is always constant.
Linear sequence: An alternative name for an arithmetic progression commonly used in Edexcel exam papers.
Common difference: The constant value added to, or subtracted from, each successive term in an arithmetic progression.
nth term: A general algebraic rule or formula used to find the value of a sequence's term at any specific position.
First term: The starting value of a sequence, represented algebraically as 'a' or 'u₁'.
Geometric progression (GP): A sequence where each term is found by multiplying the previous term by a constant, non-zero number.
Common ratio: The constant multiplier between successive terms in a geometric progression.
Surd: An irrational number expressed exactly using a root sign to avoid rounding decimals.
Exponential growth: A process where a quantity increases over time at a rate proportional to its current value, represented by a geometric sequence with a common ratio where |r| > 1.
Exponential decay: A process where a quantity decreases over time at a rate proportional to its current value, represented by a geometric sequence with a common ratio where 0 < |r| < 1.
Rationalise: The process of removing a surd or irrational number from the denominator of a fraction.
Conjugate: A binomial formed by negating the second term of a binomial, often used to rationalise denominators containing surds.

Arithmetic and Geometric Progressions

Identifying Arithmetic Progressions

The General Formula for the nth Term

u_n = a + (n - 1)d

Worked Example: Algebraic Simplification

Find the nth term of the sequence $7, 12, 17, 22 \dots$

Step 1: Identify the first term ( $a$ ) and common difference ( $d$ ). $a = 7$ , $d = 12 - 7 = 5$

Step 2: Substitute these values into the general formula. $u_n = 7 + (n - 1)5$

Step 3: Expand the brackets. $u_n = 7 + 5n - 5$

Step 4: Collect like terms to simplify. $u_n = 5n + 2$

Worked Example: Checking a Term

Is $99$ in the sequence $15, 21, 27 \dots$ ?

Step 1: Find the nth term rule using the zero term method. The common difference is $d = 6$ . The "zero term" (one step back from $15$ ) is $15 - 6 = 9$ . The sequence rule is $6n + 9$ .

Step 2: Set the rule equal to the target number. $6n + 9 = 99$

Step 3: Solve for $n$ . $6n = 90$ $n = 15$

Step 4: Interpret the result. Since $n$ is an integer, $99$ is exactly the 15th term in the sequence. If $n$ had been a decimal, the number would not be in the sequence.

Identifying Geometric Progressions

Calculating with Geometric Progressions

The Edexcel specification often represents these sequences simply as $r^n$ , but the formal general formula used to find the nth term of a geometric progression is:

u_n = ar^{n-1}

Worked Example: Finding the nth Term Expression

Find the nth term expression for a geometric progression where the second term is $-320$ and the common ratio is $-\frac{2}{5}$ .

Step 1: Find the first term ( $a$ ) using the formula $u_2 = ar$ . $-320 = a \times (-\frac{2}{5})$ $a = -320 \div (-\frac{2}{5})$ $a = 800$

Step 2: State the general nth term expression using $u_n = ar^{n-1}$ . $u_n = 800 \times (-\frac{2}{5})^{n-1}$

Worked Example: Fractional Ratio

Find the 6th term of the sequence $80, 40, 20, 10 \dots$

Step 1: Find the common ratio ( $r$ ) and first term ( $a$ ). $r = \frac{40}{80} = \frac{1}{2}$ $a = 80$

Step 2: Substitute into the formula for $n=6$ . $u_6 = 80 \times (\frac{1}{2})^{6-1}$

Step 3: Calculate the power. $u_6 = 80 \times (\frac{1}{2})^5$ $u_6 = 80 \times \frac{1}{32}$

Step 4: Find the final answer. $u_6 = \frac{80}{32} = 2.5$

Worked Example: Generating Terms with a Surd Ratio

Find the 5th term of the sequence where the nth term rule is $(\sqrt{3})^n$ .

Step 1: Substitute $n=5$ into the expression. $u_5 = (\sqrt{3})^5$

Step 2: Expand the surd multiplication. $u_5 = \sqrt{3} \times \sqrt{3} \times \sqrt{3} \times \sqrt{3} \times \sqrt{3}$

Step 3: Simplify the pairs of identical surds ( $\sqrt{3} \times \sqrt{3} = 3$ ). $u_5 = 3 \times 3 \times \sqrt{3}$

Step 4: Calculate the final exact form. $u_5 = 9\sqrt{3}$

Worked Example: Finding a Surd Common Ratio (Higher Tier)

Find the common ratio ( $r$ ) of a geometric progression where the first term is $3 + 2\sqrt{2}$ and the second term is $13 + 9\sqrt{2}$ .

Step 1: Use the ratio formula $r = \frac{u_2}{u_1}$ . $r = \frac{13 + 9\sqrt{2}}{3 + 2\sqrt{2}}$

Step 3: Expand the brackets for the numerator and denominator. Numerator: $39 - 26\sqrt{2} + 27\sqrt{2} - 36 = 3 + \sqrt{2}$ Denominator: $9 - 6\sqrt{2} + 6\sqrt{2} - 8 = 1$

Step 4: Simplify to find the exact form of $r$ . $r = 3 + \sqrt{2}$

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 use $n=0$ to try and find the first term of a sequence, but the position variable $n$ must always be a positive integer starting at $n=1$ .
2
In 'deduce the nth term' questions, examiners usually award one mark for correctly identifying the common difference (e.g., writing $4n$ ) and a second mark for finding the correct constant.
3
When a linear sequence is decreasing (e.g., $20, 15, 10 \dots$ ), ensure you include the negative sign in your nth term expression (write $-5n$ , not $5n$ ).
4
For geometric progressions, remember the power in the general formula is $n-1$ , not $n$ . Always verify your formula works by checking it against the first term.
5
If calculating a geometric sequence where the common ratio is a fraction or surd, Edexcel mark schemes frequently require the final answer to be left in exact form rather than a rounded decimal.
6
When finding the common ratio of a geometric progression containing surds (Higher Tier), you will often need to rationalise the denominator by multiplying the numerator and denominator by the conjugate.

Key Terms(12)

Arithmetic progression: A sequence where the difference between any two consecutive terms is always constant.
Linear sequence: An alternative name for an arithmetic progression commonly used in Edexcel exam papers.
Common difference: The constant value added to, or subtracted from, each successive term in an arithmetic progression.
nth term: A general algebraic rule or formula used to find the value of a sequence's term at any specific position.
First term: The starting value of a sequence, represented algebraically as 'a' or 'u₁'.
Geometric progression (GP): A sequence where each term is found by multiplying the previous term by a constant, non-zero number.
Common ratio: The constant multiplier between successive terms in a geometric progression.
Surd: An irrational number expressed exactly using a root sign to avoid rounding decimals.
Exponential growth: A process where a quantity increases over time at a rate proportional to its current value, represented by a geometric sequence with a common ratio where |r| > 1.
Exponential decay: A process where a quantity decreases over time at a rate proportional to its current value, represented by a geometric sequence with a common ratio where 0 < |r| < 1.
Rationalise: The process of removing a surd or irrational number from the denominator of a fraction.
Conjugate: A binomial formed by negating the second term of a binomial, often used to rationalise denominators containing surds.

Previous Subtopic: Term GenerationTerm Generation Next NoteQuadratic Sequences

Back to Sequence Types

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

Key Terms(12)

Arithmetic progression: A sequence where the difference between any two consecutive terms is always constant.
Linear sequence: An alternative name for an arithmetic progression commonly used in Edexcel exam papers.
Common difference: The constant value added to, or subtracted from, each successive term in an arithmetic progression.
nth term: A general algebraic rule or formula used to find the value of a sequence's term at any specific position.
First term: The starting value of a sequence, represented algebraically as 'a' or 'u₁'.
Geometric progression (GP): A sequence where each term is found by multiplying the previous term by a constant, non-zero number.
Common ratio: The constant multiplier between successive terms in a geometric progression.
Surd: An irrational number expressed exactly using a root sign to avoid rounding decimals.
Exponential growth: A process where a quantity increases over time at a rate proportional to its current value, represented by a geometric sequence with a common ratio where |r| > 1.
Exponential decay: A process where a quantity decreases over time at a rate proportional to its current value, represented by a geometric sequence with a common ratio where 0 < |r| < 1.
Rationalise: The process of removing a surd or irrational number from the denominator of a fraction.
Conjugate: A binomial formed by negating the second term of a binomial, often used to rationalise denominators containing surds.