D: Sequences and series: Use sequences and series in modelling.

The idea

This topic requires you to apply sequences and series to real-world situations. You need to recognise whether a situation is arithmetic (constant difference) or geometric (constant ratio), set up a model using standard formulae, and interpret results in context. For example, a savings plan with regular deposits may be modelled by an arithmetic series, while population growth with a fixed percentage increase is geometric. You must also consider limitations, such as discrete time steps or unrealistic long-term behaviour. The skill involves translating a verbal description into mathematical notation, solving using sum formulae, and communicating findings clearly. Assessment focuses on your ability to choose an appropriate model, carry out calculations accurately, and comment on the validity of the model in the given context.

What earns the marks

• Correctly identify whether a real-world situation should be modelled by an arithmetic or geometric sequence, giving a reason based on constant difference or constant ratio.
• Set up the general term or sum formula correctly, defining variables clearly (e.g., a for first term, d for common difference, r for common ratio, n for number of terms).
• Solve problems involving finding a specific term, the number of terms, or the sum of a sequence, showing full method.
• Interpret the mathematical solution in the context of the problem, including appropriate units and realistic constraints.
• Critically evaluate the model, discussing assumptions made and possible limitations (e.g., assuming constant growth indefinitely).

Common mistakes and how to put them right

• Confusing arithmetic and geometric sequences.

  Put it right: Check whether the change is additive or multiplicative.

• Misinterpreting the number of terms.

  Put it right: Clearly define what n represents and check with small values.

• Forgetting to convert units or round appropriately in context.

  Put it right: Relate the final answer back to the real-world quantity and round sensibly.

Exam tips

• Underline key information in the question that indicates whether the sequence is arithmetic or geometric, such as 'increases by 5 each year' or 'increases by 3% each year'.
• Write down the formula you are using before substituting values; this helps you gain method marks even if arithmetic slips occur.
• When asked to 'comment on the model', refer back to the original context and mention at least one assumption or limitation, such as ignoring external factors.