A precise definition
A sequence is an ordered list of numbers following a rule. A series is the sum of the terms of a sequence. Arithmetic sequences grow by adding a constant; geometric sequences grow by multiplying by a constant. The study of sequences and series is the mathematics of anything that occurs in discrete steps over time — loan repayments, bacterial growth, radioactive decay, stock prices, and even the convergence of algorithms.
The problem it was invented to solve
The ancient Greeks studied geometric series for philosophical reasons (Zeno's paradox: can you actually reach your destination if every step covers half the remaining distance?). The answer requires an understanding of convergent series. Medieval Indian mathematicians computed sums of arithmetic series for astronomical calculations. European mathematicians of the 17th century developed the theory of power series — infinite polynomial expressions — which became the foundation for calculus.
Where you find it in the world — including South Africa
These are not contrived textbook examples. Each application below is currently in use, driven by real institutions, and producing real outcomes.
Home loan repayments — every SA bond
Every home loan in South Africa is a geometric series problem. FNB's bond calculator, ABSA's mortgage tool, Standard Bank's home loan repayment schedule — all of them compute the sum of a geometric series to find your monthly repayment. If you understand geometric series, you can derive the formula yourself.
Compound interest and investment growth
R10,000 invested at 8% per annum compounded monthly grows as a geometric sequence. After 20 years: R49,268. With simple (arithmetic) interest: R26,000. The difference is the difference between arithmetic and geometric sequences — and why your banker will always say "start early."
Drug dosage and pharmacokinetics
When you take medication twice daily, the drug accumulates in your body as a geometric series. Each dose adds to the residual from previous doses (which decay geometrically). The 'steady-state concentration' — where accumulation and decay balance — is the sum of a convergent geometric series.
Depreciation of assets
A car that depreciates at 15% per year loses value as a geometric sequence. South Africa's Revenue Service (SARS) uses this formula for tax depreciation schedules. Businesses from Johannesburg to Cape Town compute asset values using sequences.
You've already encountered this
Every time you check your savings account and notice your interest is higher this month than last month — that's geometric growth. The '72 rule' (your money doubles every 72/interest_rate years at compound interest) is derived from geometric series. Understanding it changes how you think about money for life.
What you study — and when
- ›Arithmetic sequences: general term, sum formula
- ›Geometric sequences: general term, sum formula
- ›Sigma notation: compact representation of series
- ›Convergent geometric series: sum to infinity
- ›Financial applications: compound interest, present value, future value
- ›Annuities and loan repayments (Grade 12)
Related topics and institutions
Sequences and series are where abstract maths becomes personal finance.
The Continuum uses financial applications from the South African context to make sequences feel immediately relevant — because they are. Understanding this topic could save you hundreds of thousands of rands over a lifetime.
No card required. South African curriculum. Grade 8–12.