Financial applications of geometric sequences and series

IB Mathematics: Applications and Interpretation HL· 15 min read

1. Compound Interest and Compound Depreciation★★☆☆☆⏱ 15 min

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📘 Definition

Compound Growth

u_n = u_0 r^n

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.

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
$u_8 \approx 10000 \times 1.4221 = 14221$

2. Future Value of Ordinary Annuities★★★☆☆⏱ 20 min

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📘 Definition

Ordinary Annuity Future Value

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

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

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
$FV \approx \frac{200(1.1616 - 1)}{0.0025} \approx 12928$

3. Present Value of Annuities and Loan Repayment★★★☆☆⏱ 20 min

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📘 Definition

Ordinary Annuity Present Value

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

The current lump-sum value equivalent to a sequence of future end-of-period payments, discounted at the periodic interest rate.

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
$C \approx \frac{300000 \times 0.003333}{1 - (1.003333)^{-300}} \approx 1584$

4. Solving for Unknown Parameters★★★★☆⏱ 20 min

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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 = $2 u_{0}$ , $r = 1.05$ . 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
$n = \frac{\ln 2}{\ln(1.05)} \approx 14.2$

5. Common Pitfalls

Wrong move:

Using simple (arithmetic) growth formula instead of compound (geometric) growth

Why:

Confusion between fixed amount growth and percentage growth leads to incorrect results

Correct move:

Confirm that percentage change of current value means compound growth, which requires a geometric model

Wrong move:

Forgetting to adjust annual interest rate for non-annual compounding

Why:

Using the full annual rate directly leads to incorrectly high growth and wrong results

Correct move:

Divide annual rate by compounding periods per year, multiply years by the same number for total periods

Wrong move:

Confusing future value and present value formulas for annuities

Why:

The formulas have reversed numerators, swapping them gives nonsensical values

Correct move:

FV = future worth of past payments, PV = current worth of future payments, match the formula to the problem

Wrong move:

Rounding intermediate values too early

Why:

Small rounding errors in powers of growth factors compound to large errors in the final result

Correct move:

Keep full precision in your calculator during calculations, only round the final answer

Wrong move:

Treating an annuity due as an ordinary annuity

Why:

Payments at the start of the period earn one extra period of interest that is missed

Correct move:

Multiply the ordinary annuity result by $r$ to get the correct value for an annuity due

6. Quick Reference 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)	$F V = \frac{C ( r ^{n} - 1 )}{r - 1}$	$C$ = periodic payment, $r$ = periodic growth factor
Present value (ordinary annuity)	$P V = \frac{C ( 1 - r ^{- n} )}{r - 1}$	$P V$ = loan principal, $C$ = periodic repayment
Annuity due (start of period)	$F V_{d u e} = F V_{or d ina r y} \times r$	Add one extra period of compounding

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Compound interest investment calculation
2024 · 2
Annuity present value for mortgage
2023 · 1
Compound depreciation of vehicle

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.