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Module 7 · L5 of 20 ~40 min ⚡ +100 XP available

Geometric Sequences in Finance

Compound interest and depreciation are not new formulas, they are geometric sequences in disguise. Once you see the connection, you unlock the full power of GP sum formulas for accumulated balances and depreciation. Be careful what a SUM of balances means: adding successive snapshots of one account counts the same principal again and again, so it is a series, not a bank balance.

Today's hook, Your super fund projects a $\$50,000$ balance growing to $\$380,000$ over 30 years at 7% p.a. They didn't use magic, they used $T_{30}$ of a geometric sequence with $r = 1.07$. Every balance projection you'll ever see is a GP term.

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Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

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Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

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Think first, your gut answer

+5 XP warm-up

You invest $10,000 at 5% p.a. compound interest. The balances at the end of each year form a sequence:

$10,000 → $10,500 → $11,025 → $11,576.25 → ...

Without using a formula is this sequence arithmetic, geometric, or neither? How can you tell?

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The GP connection, one insight to rule them all

+5 XP to read Compound interest and reducing balance depreciation are geometric sequences in disguise.

The GP connection, one insight to rule them all

+5 XP to read

Compound interest and reducing balance depreciation are geometric sequences in disguise. Lock this mapping and the rest of the module is just substitution.

When you invest $P$ at rate $i$ per period, each balance is the previous balance multiplied by $(1+i)$. That constant multiplier is the common ratio of a GP, which means every compound-interest formula is just $T_n = ar^{n-1}$ in disguise.

Geometric-progression terms map directly onto compound-interest variables.

$$T_n = ar^{n-1} \quad \Longleftrightarrow \quad A = P(1+i)^n$$

Compound interest GP

$a = P(1+i)$ · $r = (1+i)$ · the rate is not $r$, the ratio is $(1+i)$.

Depreciation GP

$a = V_0(1-d)$ · $r = (1-d)$ · values shrink because $r < 1$.

$T_n$ vs $S_n$

$T_n$ = one balance. $S_n$ = sum of all balances. Don't confuse them, exam trap!

What you'll master

Know

Key facts

Compound interest balances form a GP with $r = (1+i)$
Reducing balance depreciation forms a GP with $r = (1-d)$
The GP sum formula gives the total of all balances, not the final balance

Understand

Concepts

Why the common ratio in finance is $(1+\text{rate})$, not the rate itself
What $S_n$ does and does not mean in finance: it sums balances, which is not an account total and not the interest earned
The difference between $T_n$ (one term) and $S_n$ (sum of terms)

Can do

Skills

Identify a financial scenario as a GP and state $a$ and $r$
Use $T_n$ to find any year's balance
Use $S_n$ to find total value accumulated over multiple periods

Key terms

Geometric sequence (GP)A sequence where each term is found by multiplying the previous term by a constant ratio $r$.

Common ratio $r$The constant multiplier between consecutive terms. For compound interest, $r = (1+i)$.

$T_n = ar^{n-1}$The $n$-th term formula. Gives one specific balance or value.

$S_n = \frac{a(r^n-1)}{r-1}$The sum of first $n$ terms. Gives the total of all balances accumulated.

Reducing balance depreciationA fixed percentage $d$ applied to the current value each period: $r = (1-d)$.

Principal $P$The initial amount invested or borrowed. Corresponds to $a / (1+i)$ in the GP mapping.

Compound interest as a geometric sequence

When you invest P at rate i per period, the balance at the end of each period is:

Compound interest as a geometric sequence

core concept

When you invest $P$ at rate $i$ per period, the balance at the end of each period is:

Period 1: $P(1+i)$ · Period 2: $P(1+i)^2$ · Period 3: $P(1+i)^3$ · ...

This is a geometric sequence with first term $a = P(1+i)$ and common ratio $r = (1+i)$.

$$T_n = ar^{n-1} = P(1+i) \cdot (1+i)^{n-1} = P(1+i)^n$$

Critical insight: The common ratio is not the interest rate $i$. It is $(1+i)$. Each balance is the previous balance plus the interest, you keep 100% and add the interest. If you used $r = 0.05$ for 5% interest, the sequence would shrink: $10{,}000,\ 500,\ 25,\ldots$, completely wrong.

Superannuation projections. When your super fund projects your retirement balance, it is calculating $T_n$ of a GP. $\$50,000$ today growing at 7% p.a. for 30 years is simply $T_{30}$ with $a = 50{,}000 \times 1.07$ and $r = 1.07$. That gives $\$50{,}000 \times 1.07^{30} \approx \$380{,}613$, every superannuation projection is a GP term.

Compound interest GP: $a = P(1+i)$, $r = (1+i)$, so $T_n = P(1+i)^n$ (balance after $n$ periods); Common ratio is $(1+i)$, never just $i$. Using $r = i$ will give a collapsing sequence.

Pause, copy the GP model for compound interest: first term $a = P(1+i)$, common ratio $r = (1+i)$, $n$th term $T_n = P(1+i)^n$, noting that $r = (1+i)$, not just $i$, into your book.

Quick check: An investment earns 6% p.a. compound interest. What is the common ratio $r$ of the GP formed by the year-end balances?

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · BALANCE USING $T_n$

An investment of $8,000 earns 6% p.a. compounded annually. Find the balance at the end of year 7.

$a = 8{,}000(1.06) = 8{,}480$, $\quad r = 1.06$

Identify the GP. The first term $a$ is the balance after period 1.

PROBLEM 2 · SUM FORMULA $S_n$

Find the total of all year-end balances from year 1 to year 7 for the investment above ($8,000 at 6% p.a.).

$a = 8{,}480$, $r = 1.06$, $n = 7$

Same GP as before, now we sum all 7 terms.

PROBLEM 3 · DEPRECIATION GP

A car is valued at $50,000. It depreciates at 12% p.a. reducing balance. Write the first three book values and find the value after 5 years.

$a = 50{,}000(0.88) = 44{,}000$, $\quad r = 0.88$

$d = 12\%$, so $r = 1 - 0.12 = 0.88$. First book value after 1 year.

Did you get this? True or false: $S_n$ in the GP sum formula gives the final balance after $n$ periods of compound interest.

Common errors · the 3 traps that cost marks

Trap 01

Using $r = i$ instead of $r = (1+i)$

Using the interest rate directly as the common ratio gives a collapsing sequence. Always add 1: $r = 1 + i$ for growth, $r = 1 - d$ for depreciation.

Trap 02

Confusing $T_n$ with $S_n$

$T_n$ is the single balance at period $n$. $S_n$ is the sum of all balances up to period $n$. An exam question asking "find the balance" wants $T_n$, not $S_n$.

Trap 03

Wrong power in $T_n = ar^{n-1}$

For the balance at the end of year 7, the power is $r^6$ (not $r^7$). The formula is $T_n = ar^{n-1}$. Alternatively, use $A = P(1+i)^n$ directly to avoid this confusion.

Three-things-learned: Name three things that are true about the common ratio in a compound interest GP.

Odd one out: Three of these statements about GP finance are correct. Which one is wrong?

Revisit your thinking

Quick-fire drill · 5 GP finance problems

$\$12,000$ at 6% p.a. Write $a$ and $r$ for the compound interest GP.

Use $T_4 = ar^3$ to find the balance at end of year 4 (P = $12,000, i = 6%).

$\$8,000$ at 10% p.a. At what year does the balance first exceed $\$15,000$?

Car worth $\$40,000$ depreciates at 15% p.a. Write $a$ and $r$.

Find the total of all year-end balances (years 1–5) for $\$10,000$ at 4% p.a. Use $S_5$.

Fill in the blanks: For reducing balance depreciation at rate $d$, the common ratio is r = (1 __ d), which means $r$ is __ than 1, and the sequence is __ easing (increasing / decreasing).

Revisit your thinking

The sequence $10{,}000,\ 10{,}500,\ 11{,}025,\ 11{,}576.25,\ldots$ is geometric. Test: $10{,}500/10{,}000 = 1.05$ and $11{,}025/10{,}500 = 1.05$, constant ratio. It is not arithmetic because the differences are not constant ($500,\ 525,\ 551.25,\ldots$). The ratio test is the definitive check for geometric sequences.

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Show what you have learned

Multiple choice, then short answer under exam conditions.

Short answer

ApplyBand 43 marks

Q1. An investment of $\$15,000$ earns 4% p.a. compounded annually. (a) Write the first term $a$ and common ratio $r$ for the GP of year-end balances. (b) Find the balance at the end of year 6. Show working. (3 marks)

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ApplyBand 43 marks

Q2. A piece of machinery valued at $\$25,000$ depreciates at 10% p.a. reducing balance. (a) Write $a$ and $r$. (b) Find the book value at the end of year 4. (3 marks)

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AnalyseBand 54 marks

Q3. $\$5{,}000$ is invested ONCE at 8% p.a. for 8 years. (a) The eight year-end balances form a GP. Use the GP sum formula to find $S_8$, the sum of those eight numbers, and state what that total does and does not represent. (b) Calculate the total interest the account actually earns over the 8 years. (4 marks)

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Comprehensive answers (click to reveal)

Drill 1: $a = 12{,}720$, $r = 1.06$ · 2: $T_4 = 12{,}720(1.06)^3 = \$15{,}149.72$ · 3: Year 7 (first exceeds $15,000) · 4: $a = 34{,}000$, $r = 0.85$ · 5: $a = 10{,}400$, $S_5 = 10{,}400(1.04^5-1)/0.04 = \$56{,}329$

Q1 (3 marks): (a) $a = 15{,}000(1.04) = 15{,}600$, $r = 1.04$ [1]. (b) $T_6 = 15{,}600(1.04)^5 = \$18{,}979.78$ [2].

Q2 (3 marks): (a) $a = 25{,}000(0.90) = 22{,}500$, $r = 0.90$ [1]. (b) $T_4 = 22{,}500(0.90)^3 = \$16{,}402.50$ [2].

Q3 (4 marks): (a) $a = 5{,}000(1.08) = 5{,}400$, $r = 1.08$; $S_8 = 5{,}400(1.08^8 - 1)/0.08 = \$57{,}437.79$ [1]. This is a legitimate GP sum but it is NOT money: it adds the same $\$5{,}000$ of principal in eight times over, so it is neither the account's value nor a base for interest [1]. (b) Only one deposit is ever made, so interest is the final balance minus that deposit: $5{,}000(1.08)^8 - 5{,}000 = 9{,}254.65 - 5{,}000 = \$4{,}254.65$ [2].

Consolidate and move on

Sit the module quiz, then close the lesson off.

Retrieve the lesson before you leave

State the common ratio for compound growth and for reducing-balance depreciation.
Explain how the first term changes when a sequence records year-end balances.
Decide when a finance problem needs a GP term formula and when it needs a GP sum.

Answer from memory first, then return to the matching Learn checkpoint to check and correct your response.

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A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.

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Mark lesson as complete

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Sequences and Series · Maths Advanced · Checkpoint 1