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 Video by Zeeshan Zamurred Sequences and Series

Edexcel A level Maths: 3.8 Modelling with SeriesWatch the full walkthrough 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 $fixed amount$ added each period (e.g. £500 more salary each year) is $arithmetic$ . A $fixed percentage$ change (e.g. 3% growth a year) is $geometric$ , with ratio $r = 1 + \frac{percent}{100}$ .

fixed amount \to arithmetic, fixed percent \to geometric

Spot the type from the wording.

Term vs sum

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

1Arithmetic,

a = 20000

d = 1000

n = 10

S_{10} = \frac{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 $r = 1 + \frac{p}{100}$ (or $1 - \frac{p}{100}$ for decay). Then use the geometric term or sum formula. Sum to infinity can model a long-run total when $∣ r ∣ < 1$ .

r = 1.05

; value

= 1000 \times 1.0 5^{3}

Answer£1157.63

Formula recap

arithmetic: u_{n} = a + (n - 1) d

Fixed amount each step.

geometric: u_{n} = a r^{n - 1}, r = 1 + \frac{p}{100}

Percentage change.

“in year n ” = 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

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