Modelling Geometric Growth and Decay Revision Notes for VCE SSCE General Mathematics

Modelling Geometric Growth and Decay

What is geometric growth and decay?

Geometric growth or decay occurs when a quantity increases or decreases by the same percentage at regular intervals. This creates a pattern where values change exponentially rather than by a constant amount.

For example, a sequence that starts with $3$ and doubles each time creates the pattern $3, 6, 12, 24, \ldots$ This can be written as a recurrence relation: $V_0 = 3$ , $V_{n+1} = 2V_n$ , where each term is multiplied by the same number (the common ratio).

infoNote

A recurrence relation is a formula that defines each term in a sequence based on the previous term(s). In the context of geometric growth and decay, it shows how we multiply the current value by a fixed ratio to get the next value.

Graphing geometric sequences

When we plot the terms of a geometric sequence on a graph, we can see the pattern of growth or decay visually. The graph reveals whether values are increasing exponentially (curving upward) or decreasing exponentially (curving downward and approaching zero). Let's examine two examples.

Example: Geometric growth

Consider the sequence with starting value $V_0 = 1$ and rule $V_{n+1} = 3V_n$ .

Converting to words:

Starting value $= 1$
Next value $= 3 \times$ current value

To find the first six terms, multiply each term by $3$ :

$1, 3, 9, 27, 81, 243$

When plotted, these points show exponential growth, curving upward steeply as $n$ increases.

Example: Geometric decay

Consider the sequence with starting value $V_0 = 8$ and rule $V_{n+1} = 0.5V_n$ .

Converting to words:

Starting value $= 8$
Next value $= 0.5 \times$ current value

To find the first six terms, multiply each term by $0.5$ (halving each time):

$8, 4, 2, 1, 0.5, 0.25$

When plotted, these points show exponential decay, decreasing rapidly at first then levelling off as they approach zero.

General rules for modelling geometric growth and decay

When working with geometric sequences using recurrence relations, the common ratio $R$ is the key factor that determines whether we have growth or decay.

chatImportant

Understanding the Common Ratio $R$ :

For geometric growth:

When $R > 1$ , the recurrence relation $V_{n+1} = RV_n$ models geometric growth
Values increase exponentially over time
Example: $R = 1.5$ means values multiply by $1.5$ each period

For geometric decay:

When $R < 1$ , the recurrence relation $V_{n+1} = RV_n$ models geometric decay
Values decrease exponentially over time
Example: $R = 0.8$ means values multiply by $0.8$ each period (keeping $80\%$ of the previous value)

The value of $R$ acts as a multiplier between consecutive terms in the sequence.

infoNote

Memory Aid: "R > 1 means Rising, R < 1 means Reducing"

Compound interest investments and loans

Understanding compound interest

Compound interest is more common in real-world finance than simple interest. With compound interest, any interest earned after one time period is added to the principal amount. This means the interest itself then earns interest in subsequent periods, creating geometric growth.

Consider an investment of $5000 that earns $8\%$ interest per annum, compounding yearly. The investment's value increases by $8\%$ each year.

To develop the recurrence relation:

Let $V_n$ be the value of the investment after $n$ years. Initially, $V_0 = `$ 5000$`.

To find the rule between terms:

\begin{aligned} \text{next value} &= \text{current value} + 8\% \text{ of current value} \\ &= 108\% \text{ of current value} \\ &= 1.08 \times \text{current value} \\ &= 1.08 \times V_n \end{aligned}

This gives us the recurrence relation to model the investment's growth over time.

Recurrence model for compound interest

For investments and loans that compound yearly, we use a specific formula structure that captures the growth pattern:

Let $V_n$ be the value of the investment after $n$ years.

Let $r$ be the annual percentage interest rate.

chatImportant

Compound Interest Recurrence Model:

$V_0 = \text{principal}, \quad V_{n+1} = RV_n, \quad \text{where } R = 1 + \frac{r}{100}$

Key point: Notice that $R = 1 + \frac{r}{100}$ because we keep the original amount (the " $1$ ") and ADD the interest rate as a decimal.

Memory Aid: "Compound interest: ADD the rate $(1 + r/100)$ "

lightbulbExample

Worked Example: Compound Interest Investment

Problem: The following recurrence relation models a compound interest investment of $2000 paying interest at $7.5\%$ per annum:

$V_0 = 2000, \quad V_{n+1} = 1.075 \times V_n$

where $V_n$ is the value of the investment after $n$ years.

Part a: Use the recurrence relation to show that the value after $3$ years is $2484.59.

Part b: Determine when the value will first exceed $2500.

Solution a:

Starting with the principal: $V_0 = 2000$

Calculate successive terms using the recurrence relation:

\begin{aligned} V_1 &= 1.075 \times 2000 = 2150 \\ V_2 &= 1.075 \times 2150 = 2311.25 \\ V_3 &= 1.075 \times 2311.25 = 2484.59 \end{aligned}

The value after $3$ years is $2484.59.

Solution b:

Table

Continuing the calculations:

$V_4 = 1.075 \times 2484.59375 = 2670.93828125$

The investment first exceeds $2500 after $4$ years.

Calculator tip: You can use the repeat function on your calculator. Enter $2000$ , press enter, then type $\times 1.075$ and press enter repeatedly, counting how many times until the value exceeds $2500.

Reducing balance depreciation

What is reducing balance depreciation?

Reducing balance depreciation is a method where an asset's value decreases geometrically over time. Each year, the value reduces by a fixed percentage of the previous year's value. This is different from straight-line depreciation where the same dollar amount is deducted each year.

infoNote

Reducing balance depreciation is more realistic for many assets (like vehicles or electronics) because they lose value more quickly in the early years and the rate of loss slows over time.

Recurrence model for reducing balance depreciation

For assets depreciating using the reducing balance method, we use a formula that models the decreasing value:

Let $V_n$ be the value of the asset after $n$ years.

Let $r$ be the annual percentage depreciation rate.

chatImportant

Reducing Balance Depreciation Recurrence Model:

$V_0 = \text{initial value}, \quad V_{n+1} = RV_n, \quad \text{where } R = 1 - \frac{r}{100}$

Key point: Notice that $R = 1 - \frac{r}{100}$ because we keep a portion of the value (the " $1$ ") and SUBTRACT the depreciation rate as a decimal.

Memory Aid: "Depreciation: SUBTRACT the rate $(1 - r/100)$ "

lightbulbExample

Worked Example: Setting Up a Depreciation Model

Problem: A sofa was purchased for $7500 and is depreciating at a reducing balance rate of $8.4\%$ per annum. Write a recurrence relation where $V_n$ is the value of the sofa after $n$ years.

Solution:

Identify the initial value: $V_0 = 7500$

Calculate the value of $R$ :

The depreciation rate is $8.4\%$ per annum.

$R = 1 - \frac{8.4}{100} = 0.916$

Write the recurrence relation:

$V_0 = 7500, \quad V_{n+1} = 0.916 \times V_n$

lightbulbExample

Worked Example: Using the Depreciation Model

Problem: The following recurrence relation models office furniture with a purchase price of $9600, depreciating at a reducing balance rate of $7\%$ per annum:

$V_0 = 9600, \quad V_{n+1} = 0.93 \times V_n$

where $V_n$ is the value of the office furniture after $n$ years.

Part a: Find the value of the furniture, correct to the nearest cent, after $1$ , $2$ , and $3$ years.

Part b: If the furniture was purchased in $2023$ , at the end of which year will its value first be less than $7000?

Solution a:

Starting with the purchase price: $V_0 = 9600$

Calculate successive values:

\begin{aligned} V_1 &= 0.93 \times 9600 = 8928 \\ V_2 &= 0.93 \times 8928 = 8303.04 \\ V_3 &= 0.93 \times 8303.04 = 7721.83 \end{aligned}

Solution b:

Table

Continue calculating until the value drops below $7000:

\begin{aligned} V_4 &= 0.93 \times 7721.8272 = 7181.299296 \\ V_5 &= 0.93 \times 7181.299296 = 6678.708345 \end{aligned}

The value first drops below $7000 after $5$ years. Since the furniture was purchased in $2023$ , it is first worth less than $7000 at the end of $2028$ .

bookmarkSummary

Key Points to Remember:

Geometric growth and decay occur when quantities change by the same percentage at regular intervals, creating exponential patterns.
The common ratio $R$ determines the type of change: When $R > 1$ , we have geometric growth. When $R < 1$ , we have geometric decay.
For compound interest: Use $R = 1 + \frac{r}{100}$ where you ADD the interest rate to $1$ . This models exponential growth of investments.
For reducing balance depreciation: Use $R = 1 - \frac{r}{100}$ where you SUBTRACT the depreciation rate from $1$ . This models exponential decay of asset values.
Recurrence relations provide a powerful tool for modelling real-world financial situations involving geometric growth and decay, allowing us to predict future values and answer practical questions.

Modelling Geometric Growth and Decay (VCE SSCE General Mathematics): Revision Notes