D6PureStretch

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.

30 min Sequences and Series

The big picture

A repeated-payment model depends on timing. A payment made at the start of a year earns one more year of interest than the same payment made at the end. Write the first two and last two terms before applying a geometric-series formula. Keep a clear valuation time and use a consistent interest period. A model with a fixed interest rate assumes the rate and payment schedule remain unchanged; real accounts can have fees, tax and changing rates.

Edexcel A level Maths: 3.8 Modelling with SeriesCurated by Kepler, not produced by us. Watch it, then work through 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

Payments at the beginning and end give different sums

Keep a clear valuation time and use a consistent interest period. A model with a fixed interest rate assumes the rate and payment schedule remain unchanged; real accounts can have fees, tax and changing rates.

Example 1A saver deposits 100 pounds at the end of each year. Interest is 5% per year. Find the balance immediately after the third deposit.

1The first payment earns interest for two years:

100(1.05)^2

2The second earns for one year:

100(1.05)

. The third has not yet earned interest: 100.

3Total

=100(1.1025+1.05+1)=315.25

4If the three deposits were instead at the beginnings of the years and valued at the end of year 3, every term would gain a factor 1.05.

Answer315.25 pounds under the stated end-of-year schedule.

Think like an examiner

Write the defining equations and show the substitution before evaluating.
Check the domain, units and interpretation of the final result.

Common misconceptions

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 results to remember

\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.

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.

Next step

Keep practising

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

Browse practice topics

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