Connect compound interest, geometric sequences, and exponential growth

Ontario Grade 11 Mathematics

Discrete Functions

Seeing the same pattern through three different lenses

Imagine you place CAD 1 000 in a savings account that pays 6% interest per year, compounded annually. After one year you have CAD 1 060. After two years you have CAD 1 123.60. After three years you have CAD 1 191.02. Look closely at those amounts: each one is exactly 1.06 times the amount before it. That is the same pattern you see in a geometric sequence, and it is also the defining feature of exponential growth. This lesson ties those three ideas together so you can move fluently between them.

What you will learn

Recognize that a geometric sequence and compound interest share the same multiplicative structure.
Write the terms of a geometric sequence and identify the common ratio.
Use the compound interest formula to calculate a future value after a given number of compounding periods.
Explain why repeated multiplication by a constant ratio produces exponential growth.
Connect the three representations — sequence notation, formula notation, and function notation — for the same growth pattern.

Prerequisite Bridge: Exponents and Percent Increase

Before connecting the three big ideas, let's make sure two Grade 10 tools are sharp. First, recall that increasing a quantity by a percent means multiplying it by a growth factor. A 6% increase means multiplying by

1 + 0.06 = 1.06

. A 3.5% increase means multiplying by

1.035

. The growth factor is always

1 + r

, where

r

is the interest rate written as a decimal.

Second, recall the exponent rule: multiplying the same base repeatedly gives a power. If you multiply CAD 1 000 by

1.06

three times, you get

1000 \times 1.06 \times 1.06 \times 1.06 = 1000 \times 1.06^3

. This shorthand is the bridge from repeated multiplication to exponential notation, and it is the key mechanic of this entire lesson.

A percent increase of $r$ corresponds to a multiplication factor of $1 + r$ .
Repeated multiplication by the same factor is written as a power: $a \times b^n$ .
These two facts together make the compound interest formula possible.

Geometric Sequences: Multiplication Step by Step

A sequence is simply an ordered list of numbers. A geometric sequence is one where you move from each term to the next by multiplying by the same fixed number every time. That fixed number is called the common ratio, usually written

r

. For example, the sequence

500, 600, 720, 864, \ldots

has a common ratio of

r = 1.2

because

500 \times 1.2 = 600

,

600 \times 1.2 = 720

, and so on.

The first term is written

t_1

(read 'term one'). After one multiplication you get

t_2 = t_1 \cdot r

. After two multiplications you get

t_3 = t_1 \cdot r^2

. The pattern leads to a general formula for the

n

-th term.

Notice what this formula is really saying: start at

t_1

and apply the growth factor

r

exactly

n - 1

times. This is identical to what a bank does when it applies the same interest rate once per year for several years.

t_n = t_1 · r^{n-1}

A geometric sequence multiplies by the same ratio $r$ at each step.
The general term is $t_n = t_1 \cdot r^{n-1}$ .
The common ratio can be found by dividing any term by the one before it: $r = t_{n} \div t_{n-1}$ .
If $r > 1$ , the sequence grows; if $0 < r < 1$ , it shrinks.

Compound Interest: The Same Pattern with a Financial Name

Compound interest means that interest is calculated on both the original amount and any interest already earned. The original amount deposited or borrowed is called the principal, written

P

. The interest rate per compounding period is

i

(a decimal). The number of compounding periods is

n

. The amount after

n

periods is called the future value, written

A

.

Each period, the account balance is multiplied by

(1 + i)

. After one period the balance is

P(1+i)

. After two periods it is

P(1+i)^2

. After

n

periods it is

P(1+i)^n

. Compare this to the geometric sequence formula

t_n = t_1 \cdot r^{n-1}

: if you set

t_1 = P(1+i)

(the balance after the first period) and

r = 1+i

, the two formulas describe exactly the same list of numbers.

The compound interest formula is often written as

A = P(1 + i)^n

. Here

n

counts the total number of times interest is applied. This is why

n

appears as an exponent: each compounding period is one more multiplication by the growth factor

(1+i)

.

A = P(1 + i)^n

Principal $P$ is the starting amount; future value $A$ is the ending amount.
The growth factor per period is $(1 + i)$ , exactly like the common ratio $r$ in a geometric sequence.
The compound interest formula $A = P(1+i)^n$ is a geometric sequence in disguise.
Compounding annually means one period per year; the rate $i$ equals the annual rate as a decimal.

Exponential Growth: The Function View

When the number of compounding periods

n

can be any non-negative integer, the formula

A = P(1+i)^n

defines a function. In function notation you can write

A(n) = P(1+i)^n

, where

n

is the input (time in periods) and

A(n)

is the output (account balance). This is an exponential function because the input

n

appears as an exponent.

The three representations — the sequence list, the compound interest formula, and the exponential function — all describe the same growth pattern. The sequence list shows individual terms one at a time. The formula lets you jump directly to any term without listing every one before it. The function notation makes it clear that the balance depends on time and that the relationship is exponential.

Exponential growth has a distinctive shape: it starts slow and accelerates. The balance grows by more dollars each period even though the percentage rate stays the same, because the base on which interest is calculated keeps getting larger. That is the real-world meaning of the exponent growing.

A(n) = P(1 + i)^n

Writing $A(n) = P(1+i)^n$ shows the balance as an exponential function of the number of periods.
The base of the exponential function is the growth factor $(1+i)$ , which equals the common ratio $r$ .
All three representations — sequence, formula, function — encode the same multiplicative structure.
Exponential growth accelerates over time because interest is earned on a larger and larger base.

Linking the Three Representations Side by Side

To see the connection clearly, consider a single situation: CAD 2 000 is invested at 5% per year, compounded annually. List the first few balances, then write the general term as a sequence, then write the formula, then write the function. Every representation gives the same numbers.

As a sequence:

t_1 = 2000 \times 1.05 = 2100

,

t_2 = 2100 \times 1.05 = 2205

,

t_3 = 2205 \times 1.05 = 2315.25

, and so on, with

r = 1.05

. As a compound interest formula after

n

years:

A = 2000(1.05)^n

. As a function:

A(n) = 2000(1.05)^n

. Notice that

t_1

in the sequence corresponds to

A(1)

,

t_2

to

A(2)

, and so on. The sequence index

n

and the function input

n

count the same thing: how many times the growth factor has been applied.

The key insight to carry forward is this: whenever you see a situation where a quantity is multiplied by the same factor repeatedly, you are looking at a geometric sequence, compound interest, and exponential growth all at once. The context gives it a name; the mathematics is the same.

One financial scenario can be written as a sequence list, a formula, or a function — the numbers agree.
The common ratio of the sequence equals the growth factor $(1+i)$ in the formula equals the base in the function.
Recognizing this shared structure lets you choose whichever representation is most useful for a given question.

Three Representations of the Same Growth Pattern

Representation	Notation	What n counts	Example (P = 1000, r = 1.05)
Geometric sequence	$t_n = t_1 \cdot r^{n-1}$	Term number (starting at 1)	$t_3 = 1050 \times 1.05^2 = 1157.63$
Compound interest	$A = P(1+i)^n$	Number of compounding periods	$A = 1000 \times 1.05^3 = 1157.63$
Exponential function	$A(n) = P(1+i)^n$	Input variable (time in periods)	$A(3) = 1000 \times 1.05^3 = 1157.63$

Worked example

From Sequence to Future Value

Priya opens a savings account with a deposit of CAD 3 000. The account earns 4% interest per year, compounded annually. (a) Write the first four annual balances as a geometric sequence and state the common ratio. (b) Use the compound interest formula to find the balance after 10 years. Round to the nearest cent.

Identify the growth factor
A 4% annual interest rate means each year's balance is multiplied by $1 + 0.04 = 1.04$ . This multiplier is the growth factor, and in a geometric sequence it is the common ratio $r$ .
$r = 1.04$
List the first four balances
Start with $t_1$ , the balance after year 1, then multiply by $r = 1.04$ each time to get the next term. Round intermediate values to the nearest cent for clarity.
$t_1 = 3000 × 1.04 = 3120.00$
Continue the sequence
Apply the common ratio again to get years 2, 3, and 4.
$t_2 = 3120.00 × 1.04 = 3244.80, \ t_3 = 3244.80 × 1.04 = 3374.59, \ t_4 = 3374.59 × 1.04 = 3509.57$
Write the general term
Using the geometric sequence formula $t_n = t_1 \cdot r^{n-1}$ with $t_1 = 3120$ and $r = 1.04$ , or equivalently using the compound interest formula $A = P(1+i)^n$ with $P = 3000$ , $i = 0.04$ , both correctly model the situation.
$A(n) = 3000 × (1.04)^n$
Apply the formula for n = 10
Substitute $n = 10$ into $A(n) = 3000 \times (1.04)^n$ . Calculate $(1.04)^{10}$ first: $(1.04)^{10} \approx 1.480244$ . Then multiply by 3000.
$A(10) = 3000 × (1.04)^{10} \approx 3000 × 1.480244 \approx 4440.73$

Answer: The common ratio is

r = 1.04

. The first four annual balances are approximately CAD 3 120.00, CAD 3 244.80, CAD 3 374.59, and CAD 3 509.57. After 10 years the balance is approximately CAD 4 440.73.

Check: Verify the 10-year result by checking that

3000 \times 1.04^{10}

is reasonable. Since

1.04^{10} \approx 1.48

, the balance is about 48% larger than the principal. CAD 3 000 increased by 48% gives roughly CAD 3 000 + CAD 1 440 = CAD 4 440, which matches CAD 4 440.73. The answer is consistent.

Worked example

Identifying Exponential Growth from a Sequence

A colony of bacteria doubles every hour. At time zero there are 500 bacteria. (a) Write the number of bacteria at hours 0, 1, 2, 3, and 4 as a sequence. (b) Identify the common ratio and write the count as a function of time

t

in hours. (c) Find the number of bacteria after 8 hours.

List the terms
At time zero there are 500 bacteria. Each hour the count doubles, so multiply by 2 each time.
$500, \ 1000, \ 2000, \ 4000, \ 8000$
State the common ratio
Divide any term by the previous term: $1000 \div 500 = 2$ . The common ratio is $r = 2$ , confirming this is a geometric sequence.
$r = 2$
Write the function
The starting value at $t = 0$ is 500. After $t$ hours the count has been multiplied by 2 exactly $t$ times. Write this as an exponential function using the same structure as the compound interest formula $A = P(1+i)^n$ , with $P = 500$ and the growth factor equal to 2.
$B(t) = 500 × 2^t$
Evaluate at t = 8
Substitute $t = 8$ into $B(t) = 500 \times 2^t$ . Compute $2^8 = 256$ , then multiply by 500.
$B(8) = 500 × 2^8 = 500 × 256 = 128\ 000$

Answer: The sequence for hours 0 through 4 is 500, 1 000, 2 000, 4 000, 8 000. The common ratio is

r = 2

. The function is

B(t) = 500 \times 2^t

. After 8 hours there are 128 000 bacteria.

Check: From

t = 4

(which gives 8 000) to

t = 8

is four more doublings:

8000 \times 2^4 = 8000 \times 16 = 128\ 000

. This matches

B(8) = 128\ 000

, confirming the answer is correct.

Common mistakes and how to avoid them

Using the interest rate as the growth factor instead of adding 1 first. For example, writing

A = P \times (0.06)^n

instead of

A = P \times (1.06)^n

.

Correction: Always write the growth factor as

(1 + i)

. The 1 preserves the principal; the

i

adds the interest earned.

Confusing the number of years with the number of compounding periods when interest is compounded more frequently than annually.

Correction: The exponent

n

counts compounding periods, not years. Make sure

i

and

n

use the same time unit (e.g., both monthly or both annual).

Using

t_n = t_1 \cdot r^n

instead of

t_n = t_1 \cdot r^{n-1}

, which gives an answer one multiplication too large.

Correction: The exponent is

n - 1

because

t_1

already exists before any multiplication. Multiply by

r

exactly

n - 1

times to reach term

n

.

Treating the starting value in a bacteria or population problem as

t_1

and then applying the formula as if it is the balance after one period.

Correction: When the starting value is at time zero, use

A(t) = P \cdot r^t

where the exponent equals the number of periods elapsed, not the term number minus one.

Assuming that a higher interest rate always means more total interest earned, without accounting for the number of compounding periods.

Correction: Both the rate and the number of periods affect the final amount. Use the full formula

A = P(1+i)^n

to compare scenarios fairly.

Lesson summary

A geometric sequence is a list where each term is found by multiplying the previous term by a fixed common ratio $r$ .
The general term of a geometric sequence is $t_n = t_1 \cdot r^{n-1}$ , where $t_1$ is the first term.
The compound interest formula $A = P(1+i)^n$ has the same multiplicative structure: principal $P$ acts like $t_1$ , and the growth factor $(1+i)$ acts like the common ratio $r$ .
Writing the formula as a function $A(n) = P(1+i)^n$ shows that compound interest is an exponential function of the number of periods $n$ .
All three representations — geometric sequence, compound interest formula, and exponential function — describe the same pattern of repeated multiplication by a constant factor.
Exponential growth accelerates over time because each period's interest is calculated on a larger and larger base.

Check your understanding

Question 1

A geometric sequence starts at

t_1 = 400

and has a common ratio of

r = 1.05

. Which expression gives the 6th term?

$400 \times 1.05^6$
$400 \times 1.05^5$
$400 \times 6 \times 1.05$
$400 + 5 \times 1.05$

Show answer and explanation

400 \times 1.05^5

The general term is

t_n = t_1 \cdot r^{n-1}

. For the 6th term, the exponent is

6 - 1 = 5

, giving

400 \times 1.05^5

. The other options use the wrong exponent or incorrectly add instead of multiply.

Question 2

CAD 5 000 is invested at 3% per year, compounded annually. Which formula correctly models the balance after

n

years?

$A = 5000 \times 0.03^n$
$A = 5000 \times 3^n$
$A = 5000 \times (1.03)^n$
$A = 5000 + 5000 \times 0.03 \times n$

Show answer and explanation

A = 5000 \times (1.03)^n

The growth factor for a 3% annual rate is

1 + 0.03 = 1.03

. The compound interest formula is

A = P(1+i)^n = 5000 \times (1.03)^n

. Option A uses only the rate as the base, option B confuses the percent with the base, and option D is simple interest (not compound).

Question 3

A population of insects triples every week. Starting from 200 insects at week 0, how many are there after 4 weeks?

2 400
16 200
6 400
1 800

Show answer and explanation

16 200

The function is

B(t) = 200 \times 3^t

. At

t = 4

:

3^4 = 81

, and

200 \times 81 = 16\ 200

. Option A divides incorrectly, option C uses base 2 instead of 3 (doubling, not tripling), and option D multiplies by 3 only once.

Question 4

Which statement best explains why compound interest is an example of exponential growth?

The principal increases by the same dollar amount each period.
The balance is multiplied by the same growth factor every period, so the total grows by larger and larger dollar amounts over time.
The interest rate increases each year as the balance rises.
The balance is added to itself once per year, making it grow in a straight line.

Show answer and explanation

The balance is multiplied by the same growth factor every period, so the total grows by larger and larger dollar amounts over time.

Exponential growth occurs when a quantity is multiplied by a constant factor repeatedly. In compound interest, the balance is multiplied by

(1+i)

each period. Because the base keeps growing, the dollar increase each period gets larger, even though the rate stays constant. The other options describe simple interest, increasing rates, or additive (linear) growth, none of which match compound interest.

Key terms

Geometric sequence: An ordered list of numbers where each term is found by multiplying the previous term by a fixed value called the common ratio.
Common ratio: The fixed number by which each term in a geometric sequence is multiplied to get the next term. Found by dividing any term by the term before it.
Principal: The original amount of money deposited or borrowed before any interest is added.
Compound interest: Interest calculated on both the original principal and any interest that has already been earned, causing the balance to grow faster than simple interest.
Growth factor: The number you multiply the current amount by to get the next amount. For a rate of $i$ , the growth factor is $(1 + i)$ .
Future value: The total amount in an account after a given number of compounding periods, including principal and all interest earned.
Exponential function: A function of the form $f(n) = a \cdot b^n$ , where the input $n$ appears as an exponent and $b > 0$ , $b \neq 1$ .
Compounding period: The length of time between one interest calculation and the next. Common periods include annually (once a year), semi-annually, and monthly.

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