Historical Context & Motivation
The idea of charging interest on borrowed money is one of the oldest concepts in human civilization. Ancient Mesopotamian clay tablets from around 2000 BCE already recorded loans with interest, and merchants in Babylon understood that money left to accumulate could grow substantially over time. The mathematical insight behind this growth — that each period's interest is calculated on a base that itself includes prior interest — is what we now call compound interest. This concept didn't arrive fully formed; it evolved over centuries as trade, banking, and mathematics advanced together.
On the flip side, people have always noticed that physical assets — tools, ships, buildings — lose value as they age and wear out. Accounting for this loss of value, known as depreciation, became essential for businesses and governments to track wealth accurately. Both compound interest and depreciation turn out to follow the same elegant mathematical pattern: geometric sequences.
The central question this topic addresses is: how can we predict the future value of money or assets when growth or decline happens repeatedly at a constant percentage rate? The answer lies in the powerful connection between financial applications and geometric sequences.
Core Principles & Definitions
Before diving into formulas, it's important to understand why financial calculations connect so naturally to geometric sequences. A geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a fixed value called the common ratio (r). When you invest $1,000 at 5% annual interest, after one year you have $1,000 × 1.05 = $1,050. After two years, you have $1,050 × 1.05 = $1,102.50. Each year's balance is 1.05 times the previous year's — that's a geometric sequence with r = 1.05.
Principal (P₀)
Interest / Depreciation Rate (r)
Compounding Frequency (k)
Future Value (FV)
Common Ratio
Visual Explanation
The diagram below illustrates how $1,000 grows under compound interest at 8% per year versus how a $1,000 asset depreciates at 15% per year. Notice how the compound interest curve bends upward (exponential growth) while the depreciation curve bends downward (exponential decay). Both are governed by geometric sequences — the only difference is whether the common ratio is above or below 1.
A few key observations from this graph: the compound interest curve accelerates as time goes on — it gains more dollars in year 10 than it did in year 1, because each year's interest is calculated on a larger base. Meanwhile, the depreciation curve flattens out; the asset loses less in absolute dollar terms each year, even though the percentage rate stays the same. This behavior is characteristic of geometric growth and decay, and it is fundamentally different from linear (arithmetic) change where a fixed dollar amount is added or subtracted each period.
Mathematical Framework
Let's formalize the connection between geometric sequences and financial applications. Recall that the general term of a geometric sequence is un = u1 × rn−1, where u1 is the first term and r is the common ratio. In finance, we adapt this structure to model how money grows or assets decline.
In this formula, the expression (1 + r/k) is the common ratio of the geometric sequence, and kn is the total number of compounding periods. When interest compounds annually (k = 1), the formula simplifies to FV = PV × (1 + r)n, which you can directly compare to the geometric sequence formula.
Notice the structural similarity: depreciation uses (1 − d) as its common ratio, which is always between 0 and 1, producing decay rather than growth. The IB formula booklet provides these in slightly different notation, but the underlying idea is identical. A key related concept is the geometric series, which sums multiple terms of a geometric sequence. This becomes relevant when you make regular deposits or payments.
The Effect of Compounding Frequency
One of the most important ideas in compound interest is that more frequent compounding leads to greater growth, even when the nominal annual rate stays the same. This happens because interest earned partway through the year starts earning its own interest during the remaining periods. The table below shows what happens to $1,000 invested at 12% nominal annual interest under different compounding frequencies over 5 years.
| Compounding Frequency | k (periods/year) | Rate per Period (r/k) | Total Periods (kn) | Future Value after 5 years |
|---|---|---|---|---|
| Annually | 1 | 12% = 0.12 | 5 | $1,762.34 |
| Quarterly | 4 | 3% = 0.03 | 20 | $1,806.11 |
| Monthly | 12 | 1% = 0.01 | 60 | $1,816.70 |
| Daily | 365 | 0.0329% ≈ 0.000329 | 1,825 | $1,822.03 |
As you can see, increasing the compounding frequency from annually to quarterly makes a noticeable difference ($44 more), but going from monthly to daily adds only about $5. This pattern of diminishing returns continues — there's actually a theoretical upper limit as k approaches infinity, which leads to continuous compounding using the number e. However, for IB SL 1.4, you'll typically work with finite compounding periods.
Worked Examples
Example 1: Compound Interest
Maria invests €5,000 in a savings account that pays 6% annual interest, compounded quarterly. How much will her investment be worth after 8 years?
Example 2: Depreciation
A company purchases a machine for $25,000. The machine depreciates at 20% per year. Find the value of the machine after 6 years, and determine after how many full years the machine's value first drops below $5,000.
Compound Interest vs Depreciation — Strengths & Limitations
While compound interest and depreciation are mathematically similar — both use geometric sequences — they serve very different purposes and have distinct real-world characteristics. Understanding their differences helps you recognize which formula to apply and what assumptions you're making.
| Feature | Compound Interest | Depreciation |
|---|---|---|
| Direction of change | Growth — value increases over time | Decay — value decreases over time |
| Common ratio | Greater than 1 (e.g., 1.05) | Between 0 and 1 (e.g., 0.85) |
| Compounding frequency | Can vary (annually, monthly, daily) | Typically annual in IB problems |
| Long-term behavior | Grows without bound (→ ∞) | Approaches zero but never reaches it |
| Real-world assumption | Rate stays constant (may not hold during economic crises) | Constant percentage loss (actual depreciation may be irregular) |
| Typical applications | Savings accounts, bonds, loans, mortgages | Vehicles, computers, machinery, equipment |
Connections to Advanced Topics
The financial mathematics you learn in SL 1.4 is a gateway to more sophisticated models. In higher-level courses and in real-world finance, you'll encounter extensions of these ideas that build directly on geometric sequences and series.
| SL 1.4 Concept | Advanced Extension | Where You'll See It |
|---|---|---|
| Compound interest with finite k | Continuous compounding: FV = PV × e^(rn) | IB HL, university calculus, finance |
| Single deposit | Annuities (regular payments), amortization schedules | IB HL, personal finance, actuarial science |
| Fixed depreciation rate | Straight-line depreciation, double-declining balance | Accounting, business studies |
| Geometric series (sum) | Present value of a perpetuity: S∞ = u₁/(1 − r) | IB HL, economics, investment analysis |
The concept of an infinite geometric series is particularly powerful in finance. When the common ratio satisfies |r| < 1, the sum converges to S∞ = u1/(1 − r). This is used to value assets that generate income indefinitely, like rental properties or dividend-paying stocks. You've already built the foundation — these advanced topics are natural next steps.
Practice Problems
Lesson Summary
Financial applications in IB SL 1.4 rest on one powerful idea: geometric sequences model situations where a quantity changes by a constant percentage each period. For compound interest, the formula FV = PV × (1 + r/k)kn gives the future value of an investment, where the common ratio (1 + r/k) is greater than 1, producing exponential growth. For depreciation, V = V₀ × (1 − d)n models how an asset loses value, with the common ratio (1 − d) sitting between 0 and 1, producing exponential decay.
Key skills include calculating future values, finding present values by rearranging formulas, and using logarithms to solve for time or rate. The compounding frequency affects the final value — more frequent compounding yields greater returns, though with diminishing marginal gains. These ideas extend naturally to geometric series when regular payments or deposits are involved, connecting SL 1.4 to broader topics in algebra and real-world finance.