SL 1.4 — Financial Applications of Geometric Sequences & Series
| Topic | Summary |
|---|---|
| Compound Interest | Growth model where each amount becomes the base for future growth; a geometric sequence with ratio (1 + r). |
| Depreciation | Decay model where an asset loses a fixed percentage each period; geometric decay. |
| Real vs Nominal Value | Adjust investment value for inflation to determine true purchasing power. |
| GDC Tools | Financial apps (TVM Solver) for interest, depreciation, and investment modelling. |
📌 Compound Interest as a Geometric Sequence
Compound interest is a classic example of geometric growth.
Each period multiplies the previous value by the factor (1 + r), where r is the interest rate.
- Aₙ = P(1 + r)ⁿ
- P = initial principal
- r = interest rate per period (as decimal)
- n = number of compounding periods
- This matches the structure of a geometric sequence: uₙ = a r⁽ⁿ⁻¹⁾.
Compound interest drives long-term investments, retirement funds, inflation-adjusted savings, and even national economic projections.
Example:
Invest $5,000 at 4% interest for 10 years:
A₁₀ = 5000 × (1.04)¹⁰ = $7,401.22
Always adjust both the interest rate and the
number of periods if compounding is monthly, quarterly, or half-yearly.
Failing to adjust these is a top scoring error in Paper 1 & 2.
📌 Depreciation as Geometric Decay
Depreciation reduces the value of an asset by a fixed percentage per period, creating a geometric decay pattern.
- Valueₙ = V₀ (1 − d)ⁿ
- d = depreciation rate
- The common ratio is (1 − d), always less than 1.
Example:
A car costing $30,000 depreciates by 15% annually:
Value after 4 years = 30000 × (0.85)⁴ = $18,683.06
Depreciation formulas differ across industries.
Does this imply that financial mathematics reflects constructed rather than universal truths?
How do values such as “worth” change depending on perspective?
📌 Inflation & Real Value of Investments
Inflation reduces the purchasing power of money.
To find the real value of an investment, adjust the nominal growth factor:
Real Growth Factor = (1 + r) / (1 + i)
Example:
If interest is 6% and inflation is 4%:
Real growth = 1.06 / 1.04 = 1.0192
≈ 1.92% real gain.
Economists, investors and policymakers rely on real growth rather than nominal values to compare income, GDP, and long-term investment performance.
📌 Using the GDC (TVM Solver)
The Time Value of Money (TVM) solver automates geometric financial calculations.
- N → number of periods
- I% → interest rate
- PV → present value
- FV → future value
- P/Y and C/Y → payments and compounding frequencies
Check whether your calculator is set to END mode (default).
Using BEGIN mode accidentally creates incorrect annuity models.
Always specify whether a value is nominal or
real.
Confusing these leads to incorrect final answers, especially in Paper 2 modelling questions.
📌 Applications & Modelling
This topic is widely relevant across personal finance, business investment, economics, and banking.
It forms the foundation for understanding loans, mortgages, savings schemes and asset valuations.
Investigate how the resale value of phones or laptops decays geometrically over time.
Create a model and compare predicted vs actual depreciation.
Strong IA ideas include modelling inflation-adjusted salaries, car depreciation, or comparing linear vs geometric financial models.
Include real datasets to strengthen evaluation.
An EE could examine volatility in financial models, long-term growth predictions, or geometric modelling in actuarial science.