Growth and decay; compound interest; iteration
Many real-world quantities don't grow or shrink linearly — they change by a fixed percentage each period. This is exponential growth or decay.
Compound interest
Final amount after n periods at compound rate r% per period:
A = P × (1 + r/100)ⁿ
where P = principal (initial amount), r = % rate, n = number of periods.
Worked example: £2000 invested at 4% compound interest per year for 5 years.
- A = 2000 × 1.04⁵ = 2000 × 1.21665… = £2433.31.
Depreciation (compound decrease)
A = P × (1 − r/100)ⁿ.
Worked example: a car bought at £18 000 depreciates 20% per year. Value after 3 years?
- A = 18 000 × 0.80³ = 18 000 × 0.512 = £9216.
Comparing simple and compound interest
Simple interest: I = P × r × n / 100 (interest only on original). Compound interest reinvests interest earned each period — grows faster than simple over time.
Exponential growth — populations, bacteria
Same formula. A = P × multiplierⁿ.
Worked example: a bacteria culture doubles every hour. Starting at 500 cells, find population after 8 hours.
- 500 × 2⁸ = 500 × 256 = 128 000.
Iterative processes
An iterative process repeatedly applies a formula. xₙ₊₁ = f(xₙ).
Worked example: x₁ = 3, xₙ₊₁ = (xₙ + 6/xₙ)/2 (Heron's method for √6).
- x₁ = 3.
- x₂ = (3 + 2)/2 = 2.5.
- x₃ = (2.5 + 2.4)/2 = 2.45.
- x₄ = (2.45 + 6/2.45)/2 ≈ 2.4495.
- Converging to √6 ≈ 2.4495.
Find unknowns from compound formulas
If A, P, r are known, find n by trial or logs (often trial in GCSE).
Worked example: £500 invested at 6% per year. After how many full years does it exceed £700?
- 500 × 1.06ⁿ ≥ 700.
- Trial: n = 5: 500 × 1.338 = 669.11 (no). n = 6: 500 × 1.418 = 709.26 (yes). Answer: 6 years.
⚠Common mistakes
- Using simple interest formula instead of compound — re-read the question.
- Multiplying r by n in the exponent — exponent is just n.
- Decay sign — multiplier is (1 − r/100) for decrease, not (r/100).
- Forgetting to round at end — keep exact during calculation.
- Iteration counter — x₃ uses x₂, not x₁.
➜Try this— Quick check
£3500 deposited at 2.5% compound per year for 4 years.
- A = 3500 × 1.025⁴ = 3500 × 1.10381… = £3863.34.
AI-generated · claude-opus-4-7 · v3-deep-ratio