# Financial applications of geometric sequences and series

> IB Mathematics: Applications and Interpretation HL · Number and Algebra
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u1-financial-applications-of-geometric-sequences/

This sub-topic applies geometric sequences and series to core financial problems including compound interest, asset depreciation, and regular annuity payments. You will learn to model value growth/decay and solve for unknown financial parameters.

**Prerequisites:** [Geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-hl-u1-geometric-sequences-and-series/)

## Learning objectives

- Model compound interest and depreciation using geometric sequences
- Calculate future value and present value of ordinary annuities
- Solve loan repayment problems using geometric series
- Solve for unknown time or interest rate in financial problems

## Compound Interest and Compound Depreciation

**Compound Growth** — Each period, the value is multiplied by a constant growth factor $r = 1 + i$, where $i$ is the periodic interest rate. $u_0$ = initial principal, $u_n$ = value after $n$ periods.

*Notation:* u_n = u_0 r^n

*Example:* A 5% annual interest rate gives $r = 1.05$.

Compound depreciation follows the same geometric sequence structure, but the common ratio $r < 1$, since value decreases each period. For an annual depreciation rate of $d\%$, $r = 1 - \frac{d}{100}$.

**Worked example:** An investment of \$10,000 earns 4.5% annual compound interest. What is the value after 8 years?

1. Identify parameters:
2. $$u_0 = 10000, r = 1 + 0.045 = 1.045, n = 8$$
3. Apply the geometric sequence formula for value after $n$ periods:
4. $$u_8 = u_0 r^8 = 10000 \times (1.045)^8$$
5. Calculate the result:
6. $$u_8 \approx 10000 \times 1.4221 = 14221$$

> **tip**
>
> Always check if interest is compounded more frequently than annually. For monthly compounding, divide the annual rate by 12 and use $n = 12 \times \text{years}$ for total periods.

*Calculator:* allowed

## Future Value of Ordinary Annuities

**Ordinary Annuity Future Value** — Total value of a sequence of equal end-of-period payments after $n$ periods, including accumulated compound interest. $C$ is the periodic payment.

*Notation:* FV = \frac{C\left(r^n - 1\right)}{r - 1}

*Example:* Monthly pension contributions form an ordinary annuity when contributions are made at the end of each month.

An ordinary annuity forms a finite geometric series, where the first payment earns interest for $n-1$ periods, and the final payment earns no interest. Summing this series gives the formula above.

**Worked example:** You deposit \$200 at the end of each month into an account that pays 3% annual interest compounded monthly. What is the future value after 5 years?

1. Convert to monthly parameters:
2. $$C = 200, i = \frac{0.03}{12} = 0.0025, r = 1.0025, n = 5 \times 12 = 60$$
3. Substitute into the FV formula:
4. $$FV = \frac{200\left((1.0025)^{60} - 1\right)}{1.0025 - 1}$$
5. Calculate the result:
6. $$FV \approx \frac{200(1.1616 - 1)}{0.0025} \approx 12928$$

> **exam_tip**
>
> For an annuity due (payments at the start of each period), multiply the ordinary annuity FV by an extra factor of $r$ to account for one extra period of interest.

*Calculator:* allowed

## Present Value of Annuities and Loan Repayment

**Ordinary Annuity Present Value** — The current lump-sum value equivalent to a sequence of future end-of-period payments, discounted at the periodic interest rate.

*Notation:* PV = \frac{C\left(1 - r^{-n}\right)}{r - 1}

*Example:* This formula is used to calculate the principal of a loan with fixed monthly repayments.

For a fully amortized loan (fully paid off after $n$ repayments), the present value of the repayments equals the initial loan principal. We can rearrange the formula to solve for the regular repayment amount.

**Worked example:** You take out a 25-year mortgage for \$300,000 with a 4% annual interest rate compounded monthly. What is your monthly repayment?

1. Convert to monthly parameters:
2. $$PV = 300000, i = \frac{0.04}{12} \approx 0.003333, r = 1.003333, n = 25 \times 12 = 300$$
3. Rearrange the PV formula to solve for $C$:
4. $$C = \frac{PV (r - 1)}{1 - r^{-n}}$$
5. Substitute values and calculate:
6. $$C \approx \frac{300000 \times 0.003333}{1 - (1.003333)^{-300}} \approx 1584$$

*Calculator:* allowed

## Solving for Unknown Parameters

When the number of periods $n$ or the interest rate $i$ is unknown, we use logarithms to solve for $n$, and graphing calculator solver functions to solve for $i$ as required by the IB AI HL syllabus.

**Worked example:** How long does it take an investment to double at 5% annual compound interest?

1. Let initial value = $u_0$, final value = $2u_0$, $r = 1.05$. Set up the equation:
2. $$2u_0 = u_0 (1.05)^n$$
3. Cancel $u_0$ and take natural logarithm of both sides:
4. $$\ln 2 = n \ln(1.05)$$
5. Rearrange to solve for $n$:
6. $$n = \frac{\ln 2}{\ln(1.05)} \approx 14.2$$

> **info**
>
> IB AI HL exams expect you to use your graphing calculator's equation solver to find unknown interest rates, rather than using complex algebraic methods.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using simple (arithmetic) growth formula instead of compound (geometric) growth
  - Why it fails: Confusion between fixed amount growth and percentage growth leads to incorrect results
  - Correct: Confirm that percentage change of current value means compound growth, which requires a geometric model
- **Wrong:** Forgetting to adjust annual interest rate for non-annual compounding
  - Why it fails: Using the full annual rate directly leads to incorrectly high growth and wrong results
  - Correct: Divide annual rate by compounding periods per year, multiply years by the same number for total periods
- **Wrong:** Confusing future value and present value formulas for annuities
  - Why it fails: The formulas have reversed numerators, swapping them gives nonsensical values
  - Correct: FV = future worth of past payments, PV = current worth of future payments, match the formula to the problem
- **Wrong:** Rounding intermediate values too early
  - Why it fails: Small rounding errors in powers of growth factors compound to large errors in the final result
  - Correct: Keep full precision in your calculator during calculations, only round the final answer
- **Wrong:** Treating an annuity due as an ordinary annuity
  - Why it fails: Payments at the start of the period earn one extra period of interest that is missed
  - Correct: Multiply the ordinary annuity result by $r$ to get the correct value for an annuity due

## Cheatsheet

| Scenario | Formula | Key Parameters |
| --- | --- | --- |
| Compound interest / depreciation | $u_n = u_0 r^n$ | $r=1+i$ growth, $r=1-d$ depreciation, $n$ = total periods |
| Future value (ordinary annuity) | $FV = \frac{C(r^n - 1)}{r-1}$ | $C$ = periodic payment, $r$ = periodic growth factor |
| Present value (ordinary annuity) | $PV = \frac{C(1 - r^{-n})}{r-1}$ | $PV$ = loan principal, $C$ = periodic repayment |
| Annuity due (start of period) | $FV_{due} = FV_{ordinary} \times r$ | Add one extra period of compounding |

## What's next

Modelling financial problems with geometric sequences forms a core examinable foundation for IB AI HL, and this topic regularly appears as a standalone question on both Paper 1 and Paper 2. Mastery of these formulas and adjustment for different compounding frequencies is critical to avoid losing easy marks on exam day. These concepts also underpin more advanced topics including investment optimization and statistical analysis of financial markets, connecting sequence work to the applications focus of the AI HL syllabus. You can build on this knowledge by exploring the related topics below.

- [Laws of Logarithms](https://www.owlsprep.com/study/ib-math-ai-hl-u1-laws-of-logarithms/)
- [Solving exponential equations with logarithms](https://www.owlsprep.com/study/ib-math-ai-hl-u1-solving-exponential-equations-with-logarithms/)

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