Series and Finance (HSC SSCE Mathematics Advanced): Revision Notes

Simple and Compound Interest

Introduction

This topic explores two fundamental methods of calculating interest on investments or loans: simple interest and compound interest. Understanding the difference between these is crucial for financial literacy.

Simple interest is calculated only on the original amount (the principal). It creates a linear pattern of growth and forms an arithmetic sequence when calculated over successive time periods.

Compound interest is calculated on the principal plus any accumulated interest. It creates exponential growth and forms a geometric sequence over time.

We'll also look at depreciation, which uses similar mathematics to compound interest but represents a decrease in value over time.

Simple interest

Understanding simple interest

Simple interest is interest calculated only on the original principal amount. The interest earned each period stays constant, making the total interest earned grow in a straight line over time.

The formula for simple interest is:

$I = PRn$

Where:

• $I$ is the interest earned (in dollars)
• $P$ is the principal (the original amount invested or borrowed, in dollars)
• $R$ is the interest rate per unit time (as a decimal, not a percentage)
• $n$ is the number of time periods

Connection to sequences and functions

Simple interest has interesting mathematical properties that connect it to other areas of mathematics:

As a linear function: If we treat $P$ and $R$ as fixed values, then the formula $I = PRn$ shows that interest is a linear function of time $n$. This creates a straight-line relationship when graphed.

As an arithmetic sequence: When we calculate the total interest for successive time periods ($n = 1, 2, 3, \ldots$), we get:

$T_1 = PR, \quad T_2 = 2PR, \quad T_3 = 3PR, \ldots$

This forms an arithmetic progression (AP) with:

• First term: $PR$
• Common difference: $PR$

When $n = 0$, we get $T_0 = 0$, which represents the starting point where no interest has been earned yet.

Working with simple interest

The formula $I = PRn$ gives us the interest earned only. To find the total amount at the end of $n$ time periods, we need to add the original principal back:

$\text{Total amount} = P + I = P + PRn$

This can also be written as:

$\text{Total amount} = P(1 + Rn)$

Worked example: Finding final amount and principal

Compound interest

Understanding compound interest

Compound interest is interest calculated on the principal plus any accumulated interest from previous periods. This means you earn "interest on interest", which causes the total amount to grow much faster than with simple interest.

The formula for compound interest is:

$A_n = P(1 + R)^n$

Where:

• $A_n$ is the total amount after $n$ time periods (in dollars)
• $P$ is the principal (in dollars)
• $R$ is the interest rate per unit time (as a decimal)
• $n$ is the number of time periods

Connection to sequences and functions

Compound interest also has interesting mathematical properties that reveal its exponential nature:

As an exponential function: If we treat $P$ and $R$ as fixed values, then $A_n = P(1 + R)^n$ shows that the final amount is an exponential function of time $n$ with base $(1 + R)$. This creates a curved, upward-sloping relationship when graphed.

As a geometric sequence: When we calculate the final amounts after successive time periods ($n = 1, 2, 3, \ldots$), we get:

$A_1 = P(1 + R), \quad A_2 = P(1 + R)^2, \quad A_3 = P(1 + R)^3, \ldots$

This forms a geometric progression (GP) with:

• First term: $P(1 + R)$
• Common ratio: $(1 + R)$

When $n = 0$, we get $A_0 = P$, which represents the starting amount equal to the principal.

How compound interest works: The proof

Understanding how compound interest builds up helps explain why the formula works. Let's trace through the calculation step by step.

Start with a principal $P$ and an interest rate $R$ per unit time.

After 1 time period:

$A_1 = \text{principal} + \text{interest}$

$A_1 = P + PR$

$A_1 = P(1 + R)$

This shows that adding the interest is the same as multiplying by $(1 + R)$.

After 2 time periods:

We multiply the amount after 1 period by $(1 + R)$:

$A_2 = A_1(1 + R)$

$A_2 = P(1 + R) \times (1 + R)$

$A_2 = P(1 + R)^2$

After 3 time periods:

$A_3 = A_2(1 + R)$

$A_3 = P(1 + R)^2 \times (1 + R)$

$A_3 = P(1 + R)^3$

After 4 time periods:

$A_4 = A_3(1 + R)$

$A_4 = P(1 + R)^3 \times (1 + R)$

$A_4 = P(1 + R)^4$

This pattern continues, so after $n$ time periods:

$A_n = A_{n-1}(1 + R)$

$A_n = P(1 + R)^n$

Working with compound interest

Understanding the different types of calculations you might need to perform:

Finding the total amount: Use the formula $A_n = P(1 + R)^n$ directly.

Finding the interest earned: Subtract the original principal from the final amount:

$\text{Interest} = A_n - P$

Finding the time period: Sometimes you need to find how long it takes for an investment to reach a certain amount. This requires using logarithms to solve exponential equations.

Worked example: Compound interest calculations

Depreciation

Understanding depreciation

When a business buys equipment, that equipment becomes worn or obsolete over time and loses value. Depreciation is the accounting term for this loss of value. Companies must record depreciation as an expense, which reduces their profit and therefore their income tax.

Depreciation is usually expressed as the loss per time period of a percentage of the current value. The formula for depreciation is the same as compound interest, except the rate is negative (representing a decrease rather than an increase).

The formula for depreciation is:

$A_n = P(1 - R)^n$

Where:

• $A_n$ is the value after $n$ time periods (in dollars)
• $P$ is the original cost (in dollars)
• $R$ is the depreciation rate per unit time (as a positive decimal, even though it represents a decrease)
• $n$ is the number of time periods

As an exponential function: The depreciated value is an exponential function of time $n$ with base $(1 - R)$. Because $(1 - R)$ is less than 1, the value decreases over time.

As a geometric sequence: Substituting $n = 1, 2, 3, \ldots$ gives a GP with:

• First term: $P(1 - R)$
• Common ratio: $(1 - R)$ (less than 1, so the sequence decreases)

When $n = 0$, we get $A_0 = P$, which represents the original value before any depreciation.

Working with depreciation

Understanding the calculations you might need to perform:

Finding the depreciated value: Use the formula $A_n = P(1 - R)^n$ directly.

Finding the loss of value: Subtract the depreciated value from the original cost:

$\text{Loss of value} = P - A_n$

Finding the time period: Use logarithms to solve exponential equations, similar to compound interest problems.

Worked example: Depreciation calculations

Remember!