3.8PureStretch

Modelling with Series

Sequences and series model real situations: a fixed yearly pay rise is arithmetic, while interest or population growth by a percentage is geometric. The skill is recognising which type fits and applying the right formula.

25 min Sequences and Series

Edexcel A level Maths: 3.8 Modelling with SeriesCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Decide whether a context is arithmetic or geometric
Set up the model with a, d or r
Use the term and sum formulas in context
Interpret the answer in the real situation

Which model?

A $\textbf{fixed amount}$ added each period (e.g. £500 more salary each year) is $\textbf{arithmetic}$ . A $\textbf{fixed percentage}$ change (e.g. 3% growth a year) is $\textbf{geometric}$ , with ratio $r = 1 + \tfrac{\text{percent}}{100}$ .

\text{fixed amount} \to \text{arithmetic}, \quad \text{fixed percent} \to \text{geometric}

Spot the type from the wording.

Term vs sum

“Value in year $n$ ” needs the $\textbf{nth term}$ ; “total over $n$ years” needs the $\textbf{sum}$ . Reading the question for “in the nth” versus “total/altogether” tells you which.

Example 1Salary starts at £20,000, rising £1,000 a year. Total earned over 10 years?

1Arithmetic,

a = 20000

d = 1000

n = 10

S_{10} = \tfrac{10}{2}(2(20000) + 9(1000))

Answer

£245,000

Tip — “In year n” → nth term; “total over n years” → sum Sₙ.

Percentage growth

For $p\%$ growth, the multiplier each period is $r = 1 + \tfrac{p}{100}$ (or $1 - \tfrac{p}{100}$ for decay). Then use the geometric term or sum formula. Sum to infinity can model a long-run total when $|r| < 1$ .

Example 1An investment of £1000 grows 5% per year. Value after 3 years?

r = 1.05

; value

= 1000 \times 1.05^3

Answer

£1157.63

Formula recap

\text{arithmetic: } u_n = a + (n-1)d

Fixed amount each step.

\text{geometric: } u_n = ar^{n-1}, \ r = 1 + \tfrac{p}{100}

Percentage change.

\text{“in year } n\text{” = term; “total” = sum}

Reading the question.

Common mistakes to avoid

Using an arithmetic model for percentage growth.

A fixed percentage change is geometric (multiply by r).

Giving a term when the total is asked (or vice versa).

“In the nth year” = term; “total/altogether” = sum.

Key takeaways

Fixed amount each period → arithmetic; fixed percentage → geometric (r = 1 + p/100).
“Value in year n” → nth term; “total over n years” → sum.
Use sum to infinity for a long-run total when |r| < 1.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

A salary rising by a fixed £500 each year is modelled as:
- Arithmetic
- Geometric
- Periodic
- Quadratic
Show answer and explanation
Answer: Arithmetic
Fixed amount added.
Growth of 5% per year is modelled with ratio:
- $1.05$
- $0.05$
- $5$
- $0.95$
Show answer and explanation
Answer: $1.05$
r = 1 + 5/100.
“Total earned over 10 years” requires the:
- Sum Sₙ
- nth term
- Common ratio
- Sum to infinity
Show answer and explanation
Answer: Sum Sₙ
Total = sum.
Salary £20,000 + £1,000/yr; total over 10 years:
- £245,000
- £200,000
- £290,000
- £29,000
Show answer and explanation
Answer: £245,000
5(40000 + 9000) = 245,000.
£1000 growing 5%/yr is worth, after 3 years, about:
- £1157.63
- £1150
- £1015
- £3150
Show answer and explanation
Answer: £1157.63
1000 × 1.05³.
“The amount in the nth year” requires the:
- nth term
- Sum
- Ratio
- Difference
Show answer and explanation
Answer: nth term
A single year’s value.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.