1.7.6 Sequences and series in modelling (A-level only)

What you'll learn

How to recognise situations modelled by arithmetic or geometric sequences.
How to use finite and infinite series to calculate totals.
How recurrence relations model quantities that change step by step.
How to interpret, check and criticise a mathematical model.

Sequences, series and models

A sequence is an ordered list of numbers. Each number is called a term, and terms are usually labelled $u1,u2,u3,…u_1,u_2,u_3,\ldots$

A series is the result of adding the terms of a sequence:

u1+u2+u3+⋯u_1+u_2+u_3+\cdots

A mathematical model uses mathematics to represent a real situation. In this topic, sequences describe quantities at separate times or stages, while series describe accumulated totals.

For example, yearly salaries form a sequence. The total salary earned over several years is a series.

Definition

The index

The index $n$ identifies a term's position. You must decide what $n = 0$ or $n = 1$ represents in the context, such as the purchase date or the end of the first year.

Common Mistake

Starting at the wrong term

Do not assume that the initial value is always $u_1$ . A model may use $u_0$ for the value at time zero, before any change has occurred.

Arithmetic models

An arithmetic sequence changes by the same amount at every stage. This fixed amount is called the common difference, denoted by $d$ .

If the first term is $a$ , then:

u_n=a+(n-1)d

The sum of the first $n$ terms is:

Sn=n2(2a+(n−1)d)S_n=\frac{n}{2}\left(2a+(n-1)d\right)

Equivalently, if the final term is $l$ :

Sn=n2(a+l)S_n=\frac{n}{2}(a+l)

Arithmetic models are suitable when a quantity increases or decreases by a constant amount, such as a salary rising by £1200 each year.

Key Idea

Recognising an arithmetic model

Words such as “increases by”, “decreases by” or “an extra fixed amount each year” usually indicate a constant difference.

Example

Modelling annual production

A factory produces 480 components in its first year. Production increases by 35 components each year. Find the production in year 8 and the total production over the first 8 years.

The yearly increase is constant, so use an arithmetic sequence with $a = 480$ and $d = 35$ .
Substitute $n = 8$ into the term formula:
$u_8=480+(8-1)(35)=725$
The factory produces 725 components in year 8.
Use the finite-series formula to find the accumulated production:
$S8=82(480+725)=4820S_8=\frac{8}{2}(480+725)=4820$
The total production over the first 8 years is 4820 components.

Geometric models

A geometric sequence changes by the same multiplying factor at each stage. This factor is called the common ratio, denoted by $r$ .

If the first term is $a$ , then:

u_n=ar^{n-1}

The sum of the first $n$ terms is:

Sn=a(1−rn)1−r,r≠1S_n=\frac{a(1-r^n)}{1-r},\qquad r\neq 1

Geometric models are commonly used for compound interest, population growth, depreciation and radioactive decay.

A percentage multiplier is found by converting the percentage change into a decimal:

an increase of 6% gives multiplier 1.06;
a decrease of 6% gives multiplier 0.94.

Key Idea

Recognising a geometric model

A constant percentage change produces a constant multiplier, so it should be modelled geometrically rather than arithmetically.

Example

Modelling depreciation

A car is bought for £24,000 and loses 18% of its value each year. Find its modelled value after 5 years.

The car retains 82% of its value each year, so the multiplier is $r = 0.82$ .
Let $V_n$ be the value after $n$ complete years. Since £24,000 is the value at the purchase date, use $V_0=24000$ :
$V_n=24000(0.82)^n$
Substitute $n = 5$ :
$V_5=24000(0.82)^5=8897.7562368$
The modelled value is approximately £8897.76.

Tip

Check the direction

For growth, the multiplier should be greater than 1. For decay or depreciation, it should be between 0 and 1.

Infinite geometric series

An infinite series continues without end. An infinite geometric series has a finite sum only when:

∣ r ∣ < 1

In this case, the terms approach zero and:

S∞=a1−rS_\infty=\frac{a}{1-r}

This does not mean that the sequence actually stops. It means that its accumulated total approaches a fixed limiting value.

Common Mistake

When the infinite sum does not exist

If $∣r∣≥1|r|\geq 1$ , the terms do not decrease towards zero, so you must not use $S∞=a1−rS_\infty=\frac{a}{1-r}$ .

Example

Calculating the distance travelled by a bouncing ball

A ball is dropped from a height of 2 m. After each bounce, it rises to 75% of its previous height. Find the total vertical distance predicted by the model.

The initial drop contributes 2 m. The rebound heights form a geometric sequence with first term $2 (0.75) = 1.5$ and ratio $r = 0.75$ .
Because $∣0.75∣ < 1$ , the total of all rebound heights is:
$1.51−0.75=6\frac{1.5}{1-0.75}=6$
Each rebound height is travelled twice: once upwards and once downwards. Including the initial drop gives:
$2 + 2 (6) = 14$
The model predicts a total vertical distance of 14 m.

Common Mistake

Forgetting the initial movement

In a bouncing-ball model, the first drop occurs only once, while each later rebound height normally contributes both an upward and a downward journey.

Recurrence relations

A recurrence relation defines each term using one or more earlier terms. It must be given with an initial value so that the sequence can be generated.

For example:

Pn+1=1.12Pn+50,P0=800P_{n+1}=1.12P_n+50,\qquad P_0=800

This could model a population that grows by 12% each year and then receives 50 additional individuals.

The order of operations matters. The model above applies the percentage growth first and adds 50 afterwards.

Example

Using a population recurrence model

A population is modelled by $P_{n+1}=1.12P_n+50$ , where $P_0=800$ . Find the modelled population after 2 years.

Use $P_0=800$ to calculate the value after the first year:
$P_1=1.12(800)+50=946$
Apply the same recurrence to the new value:
$P_2=1.12(946)+50=1109.52$
The model predicts approximately 1110 individuals after 2 years. Keeping the unrounded value during the calculation prevents unnecessary rounding error.

Building and evaluating a model

A sequence model normally follows this process:

Define the variable and say what its index represents.
Identify whether the change is additive, multiplicative or given recursively.
Write the sequence, term formula, series or recurrence relation.
calculate the required term or accumulated total.
Interpret the answer in context, including sensible units and rounding.
Consider whether the assumptions remain realistic.

Models often assume that a difference, percentage rate or ratio remains constant. In reality, prices, populations and production levels may be affected by changing conditions.

Key Idea

A model is an approximation

A mathematically correct answer may still be unrealistic if the model is used too far beyond the data or if its assumptions no longer apply.

Example

Assessing a long-term growth prediction

A town's population is modelled as increasing by 4% every year. Explain why the model may become unreliable over a long period.

A 4% annual increase gives a geometric model, so the predicted population grows by the multiplier 1.04 each year.
The model assumes that the percentage growth rate remains unchanged indefinitely.
Over a long period, limits such as housing, employment and available land may reduce growth. The model may therefore be useful for short-term prediction but unsuitable for distant forecasts.

Exam technique

In the exam

Define what $n$ represents and check whether the initial value corresponds to $n = 0$ or $n = 1$ .
Decide whether the situation has a constant difference, a constant ratio or a recurrence rule.
Distinguish between finding one term and finding the sum of several terms.
For an infinite geometric series, verify that $∣ r ∣ < 1$ before using the sum formula.
Interpret your answer in context, round only when appropriate, and comment on the model's assumptions if asked.

Self review

Check yourself

How can you tell whether a percentage-change model is arithmetic or geometric?
Why is the initial drop treated separately in a bouncing-ball model?
What assumptions might make a sequence model unreliable over a long period?

What you'll learn

How to recognise situations modelled by arithmetic or geometric sequences.
How to use finite and infinite series to calculate totals.
How recurrence relations model quantities that change step by step.
How to interpret, check and criticise a mathematical model.

Sequences, series and models

A sequence is an ordered list of numbers. Each number is called a term, and terms are usually labelled $u1,u2,u3,…u_1,u_2,u_3,\ldots$

A series is the result of adding the terms of a sequence:

u1+u2+u3+⋯u_1+u_2+u_3+\cdots

A mathematical model uses mathematics to represent a real situation. In this topic, sequences describe quantities at separate times or stages, while series describe accumulated totals.

For example, yearly salaries form a sequence. The total salary earned over several years is a series.

Definition

The index

The index $n$ identifies a term's position. You must decide what $n = 0$ or $n = 1$ represents in the context, such as the purchase date or the end of the first year.

Common Mistake

Starting at the wrong term

Do not assume that the initial value is always $u_1$ . A model may use $u_0$ for the value at time zero, before any change has occurred.

Arithmetic models

An arithmetic sequence changes by the same amount at every stage. This fixed amount is called the common difference, denoted by $d$ .

If the first term is $a$ , then:

u_n=a+(n-1)d

The sum of the first $n$ terms is:

Sn=n2(2a+(n−1)d)S_n=\frac{n}{2}\left(2a+(n-1)d\right)

Equivalently, if the final term is $l$ :

Sn=n2(a+l)S_n=\frac{n}{2}(a+l)

Arithmetic models are suitable when a quantity increases or decreases by a constant amount, such as a salary rising by £1200 each year.

Key Idea

Recognising an arithmetic model

Words such as “increases by”, “decreases by” or “an extra fixed amount each year” usually indicate a constant difference.

Example

Modelling annual production

A factory produces 480 components in its first year. Production increases by 35 components each year. Find the production in year 8 and the total production over the first 8 years.

The yearly increase is constant, so use an arithmetic sequence with $a = 480$ and $d = 35$ .
Substitute $n = 8$ into the term formula:
$u_8=480+(8-1)(35)=725$
The factory produces 725 components in year 8.
Use the finite-series formula to find the accumulated production:
$S8=82(480+725)=4820S_8=\frac{8}{2}(480+725)=4820$
The total production over the first 8 years is 4820 components.

Geometric models

A geometric sequence changes by the same multiplying factor at each stage. This factor is called the common ratio, denoted by $r$ .

If the first term is $a$ , then:

u_n=ar^{n-1}

The sum of the first $n$ terms is:

Sn=a(1−rn)1−r,r≠1S_n=\frac{a(1-r^n)}{1-r},\qquad r\neq 1

Geometric models are commonly used for compound interest, population growth, depreciation and radioactive decay.

A percentage multiplier is found by converting the percentage change into a decimal:

an increase of 6% gives multiplier 1.06;
a decrease of 6% gives multiplier 0.94.

Key Idea

Recognising a geometric model

A constant percentage change produces a constant multiplier, so it should be modelled geometrically rather than arithmetically.

Example

Modelling depreciation

A car is bought for £24,000 and loses 18% of its value each year. Find its modelled value after 5 years.

The car retains 82% of its value each year, so the multiplier is $r = 0.82$ .
Let $V_n$ be the value after $n$ complete years. Since £24,000 is the value at the purchase date, use $V_0=24000$ :
$V_n=24000(0.82)^n$
Substitute $n = 5$ :
$V_5=24000(0.82)^5=8897.7562368$
The modelled value is approximately £8897.76.

Tip

Check the direction

For growth, the multiplier should be greater than 1. For decay or depreciation, it should be between 0 and 1.

Infinite geometric series

An infinite series continues without end. An infinite geometric series has a finite sum only when:

∣ r ∣ < 1

In this case, the terms approach zero and:

S∞=a1−rS_\infty=\frac{a}{1-r}

This does not mean that the sequence actually stops. It means that its accumulated total approaches a fixed limiting value.

Common Mistake

When the infinite sum does not exist

If $∣r∣≥1|r|\geq 1$ , the terms do not decrease towards zero, so you must not use $S∞=a1−rS_\infty=\frac{a}{1-r}$ .

Example

Calculating the distance travelled by a bouncing ball

A ball is dropped from a height of 2 m. After each bounce, it rises to 75% of its previous height. Find the total vertical distance predicted by the model.

The initial drop contributes 2 m. The rebound heights form a geometric sequence with first term $2 (0.75) = 1.5$ and ratio $r = 0.75$ .
Because $∣0.75∣ < 1$ , the total of all rebound heights is:
$1.51−0.75=6\frac{1.5}{1-0.75}=6$
Each rebound height is travelled twice: once upwards and once downwards. Including the initial drop gives:
$2 + 2 (6) = 14$
The model predicts a total vertical distance of 14 m.

Common Mistake

Forgetting the initial movement

In a bouncing-ball model, the first drop occurs only once, while each later rebound height normally contributes both an upward and a downward journey.

Recurrence relations

A recurrence relation defines each term using one or more earlier terms. It must be given with an initial value so that the sequence can be generated.

For example:

Pn+1=1.12Pn+50,P0=800P_{n+1}=1.12P_n+50,\qquad P_0=800

This could model a population that grows by 12% each year and then receives 50 additional individuals.

The order of operations matters. The model above applies the percentage growth first and adds 50 afterwards.

Example

Using a population recurrence model

A population is modelled by $P_{n+1}=1.12P_n+50$ , where $P_0=800$ . Find the modelled population after 2 years.

Use $P_0=800$ to calculate the value after the first year:
$P_1=1.12(800)+50=946$
Apply the same recurrence to the new value:
$P_2=1.12(946)+50=1109.52$
The model predicts approximately 1110 individuals after 2 years. Keeping the unrounded value during the calculation prevents unnecessary rounding error.

Building and evaluating a model

A sequence model normally follows this process:

Define the variable and say what its index represents.
Identify whether the change is additive, multiplicative or given recursively.
Write the sequence, term formula, series or recurrence relation.
calculate the required term or accumulated total.
Interpret the answer in context, including sensible units and rounding.
Consider whether the assumptions remain realistic.

Models often assume that a difference, percentage rate or ratio remains constant. In reality, prices, populations and production levels may be affected by changing conditions.

Key Idea

A model is an approximation

A mathematically correct answer may still be unrealistic if the model is used too far beyond the data or if its assumptions no longer apply.

Example

Assessing a long-term growth prediction

A town's population is modelled as increasing by 4% every year. Explain why the model may become unreliable over a long period.

A 4% annual increase gives a geometric model, so the predicted population grows by the multiplier 1.04 each year.
The model assumes that the percentage growth rate remains unchanged indefinitely.
Over a long period, limits such as housing, employment and available land may reduce growth. The model may therefore be useful for short-term prediction but unsuitable for distant forecasts.

Exam technique

In the exam

Define what $n$ represents and check whether the initial value corresponds to $n = 0$ or $n = 1$ .
Decide whether the situation has a constant difference, a constant ratio or a recurrence rule.
Distinguish between finding one term and finding the sum of several terms.
For an infinite geometric series, verify that $∣ r ∣ < 1$ before using the sum formula.
Interpret your answer in context, round only when appropriate, and comment on the model's assumptions if asked.

Self review

Check yourself

How can you tell whether a percentage-change model is arithmetic or geometric?
Why is the initial drop treated separately in a bouncing-ball model?
What assumptions might make a sequence model unreliable over a long period?

What you'll learn

Sequences, series and models

Arithmetic models

Geometric models

Infinite geometric series

Recurrence relations

Building and evaluating a model

Review 1.7.6 Sequences and series in modelling (A-level only) by teaching Genie

1.7.6 Sequences and series in modelling (A-level only)

What you'll learn

Sequences, series and models

Arithmetic models

Geometric models

Infinite geometric series

Recurrence relations

Building and evaluating a model

Review 1.7.6 Sequences and series in modelling (A-level only) by teaching Genie

1.7.6 Sequences and series in modelling (A-level only)

What you'll learn

Sequences, series and models

Arithmetic models

Geometric models

Infinite geometric series

Recurrence relations

Building and evaluating a model

Review 1.7.6 Sequences and series in modelling (A-level only) by teaching Genie

1.7.6 Sequences and series in modelling (A-level only) Revision Guide

1.7.6 Sequences and series in modelling (A-level only)

What you'll learn

Sequences, series and models

Arithmetic models

Geometric models

Infinite geometric series

Recurrence relations

Building and evaluating a model

Review 1.7.6 Sequences and series in modelling (A-level only) by teaching Genie

1.7.6 Sequences and series in modelling (A-level only) Revision Guide