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Modelling with sequences

Model fixed changes, percentage growth and decay with sequences. Choose time indices, check integer thresholds, fit a ratio and handle caps with original worked practice.

Before you startArithmetic and geometric sequences, logarithms and inequalities

01 / Choose the model

Translate the change in words into an operation.

A sequence model describes a quantity at separate stages: the end of each month, the start of each year or after each machine cycle. Define exactly what a term measures before choosing a formula.

Fixed amount added each step → arithmetic
Fixed percentage change each step → geometric

Adding 40 units each month and increasing by 8% each month both add 40 in the first month when the initial amount is 500. They then separate: a percentage acts on the new amount at each step.

Fixed amount or fixed percentage?Explore

Dots represent month-end observations, including the starting value at t = 0. The scale is the same for every rule.

t = 6: 500 + 40 × 6 = 740 units.

Change during month 6: +40 units.

Monthly change Elapsed months 6

Watch elapsed time and the number of changes

Pause, replay or seek freely. The notes explain the same idea and stay in view.

02 / Time zero and year one

Count changes, not labels.

A stock starts at 500 units and gains 40 each monthWorked example

Q₀ = 500; Qₜ = 500 + 40t

Here t counts completed months. At t = 0 no increase has occurred.

Q₆ = 500 + 40(6) = 740

Six months means six increases.

A workshop makes 240 items in year 1 and 35 more each following yearWorked example

uₙ = 240 + 35(n − 1)

Year 1 contains the first amount. By year n there have been n − 1 increases.

u₄ = 345

Year 4 follows three increases, not four.

These are amounts at individual stages. Adding all the yearly outputs is a series calculation, covered in the next lesson.

03 / Percentage growth and decay

Use a multiplier for each full time interval.

Increase by p%: multiply by 1 + p/100
Decrease by p%: multiply by 1 − p/100

A device retains 85% of its value each yearWorked example

V₀ = 800; Vₜ = 800(0.85)ᵗ

A 15% fall leaves 85%, so the multiplier is 0.85.

V₃ = 491.30

This is the modelled value after three complete years, in pounds.

Do not subtract 15% of the original value every year: that describes a fixed annual loss. Keep unrounded values during working and round the requested final result. A model need not include the annual penny rounding of a real account unless the question specifies it.

04 / First whole period to reach a target

Use logarithms, then check the neighbouring integers.

An initial quantity of 1200 grows by 6% per year. When does it first reach 2400?Worked example

1200(1.06)ᵗ ≥ 2400, so (1.06)ᵗ ≥ 2

The index t counts whole years after the starting observation.

t ≥ ln 2 / ln 1.06 ≈ 11.8957

ln 1.06 is positive, so division preserves the inequality.

t = 12 is the first possible whole year

At t = 11 the multiplier is about 1.8983; at t = 12 it is about 2.0122.

When does 800(0.85)ᵗ first fall below 400?Worked example

(0.85)ᵗ < 0.5

The target is half of the starting value.

t > ln 0.5 / ln 0.85 ≈ 4.2650

ln 0.85 is negative, so the inequality reverses.

The first whole year is t = 5

At t = 4 the value is 417.605; at t = 5 it is 354.96425.

For “at least”, equality qualifies. For “more than” or “below”, it does not. If the logarithmic boundary is an integer, a strict inequality needs the next integer. Do not round a threshold to the nearest integer.

05 / Find a rate from observations

The gap between observations is the number of multiplications.

A positive quantity is 720 at month 2 and 1125 at month 4Worked example

1125/720 = r² = 25/16

There are two monthly intervals between the readings.

r = 5/4 = 1.25

Use the positive root because the modelled quantity remains positive.

The monthly increase is 25%

The ratio is not itself the percentage increase.

Q₀ = 720/(1.25)² = 460.8

Work backwards two months to obtain the initial value.

The observations alone do not prove that a constant percentage model will keep working. A fitted rate is conditional on the geometric assumption.

06 / Match the time unit

A monthly or quarterly rate cannot be used as a yearly rate unchanged.

A quantity grows by 2% each quarterWorked example

Annual multiplier = (1.02)⁴ = 1.08243216

Four complete quarterly multiplications occur in a year.

Annual increase = 8.243216%

Subtract 1 before converting the multiplier to a percentage.

Multiplying the rate by four gives only an approximation. Conversely, a positive annual multiplier R corresponds to quarterly multiplier R^(¼) if the same growth factor applies each quarter.

Values between the observation times require an extra assumption. A discrete annual rule by itself does not establish what happens halfway through a year.

07 / When growth stops at a cap

State the piecewise rule and check the first capped stage.

Output starts at 240 items in year 1, rises by 35 a year and is capped at 500Worked example

uₙ = min(240 + 35(n − 1), 500)

The cap prevents an overshoot: take the smaller of the proposed output and 500.

240 + 35(n − 1) ≥ 500 gives n ≥ 59/7

The first capped year is n = 9.

u₈ = 485 and u₉ = 500

The uncapped prediction for year 9 would be 520.

For n ≥ 9, uₙ = 500

The entire sequence is not arithmetic because the differences change at the cap.

“Stop increasing before an increase would exceed 500” is a different rule: it could stay at 485. Use the rule actually given rather than silently choosing how an overshoot is handled.

08 / Explain limitations in context

A mathematical model is an assumption that can be tested.

A production model may fail because staff, demand or storage capacity change. A constant percentage model for a stock may ignore additions, removals or a changing rate. Name the mechanism that breaks the assumption rather than only saying “it is unrealistic”.

Not every recurrence is arithmetic or geometricWorked example

Qₜ₊₁ = 0.9Qₜ + 20 with Q₀ = 100

Ten percent is lost and then 20 units are added.

Q₁ = 110, Q₂ = 119, Q₃ = 127.1

Neither the differences nor the ratios are constant.

A factory predicts production will rise by 12% every year indefinitely. Give a specific limitation.

Hint

Identify a real constraint ignored by the model.

Worked solution

For example, its machines have finite capacity, so indefinite percentage growth would eventually exceed the maximum output possible without expansion. Other relevant, explained limitations are valid.

10 / Recap

A useful answer includes its assumptions and units.

Define the quantity and starting index.
Distinguish an added amount from a percentage change.
Match the multiplier to the time unit.
Use logarithms for thresholds and check adjacent integers.
State how caps and partial intervals are treated.
Explain a limitation tied to the real situation.

Back to sequences and series →

Section 1 of 10 · Choose the model