Connect ordinary simple annuities with geometric series

Ontario Grade 11 Mathematics

Discrete Functions

How repeated equal payments grow into a geometric series — and how to find their total value

Suppose you deposit CAD 200 into a savings account at the end of every month, and the account earns compound interest. After a year, your balance is not simply 12 × CAD 200, because each deposit earns a different amount of interest depending on how long it has been sitting in the account. The surprising result is that the total balance turns out to be the sum of a geometric series — the same kind of series you studied when learning about sequences. This lesson builds that connection step by step, starting from what you already know about geometric sequences and compound interest.

What you will learn

Explain what an ordinary simple annuity is and identify its key features.
Recognize that the future value of an annuity is the sum of a geometric series.
State the first term, common ratio, and number of terms of that geometric series.
Apply the geometric series sum formula to calculate the future value of an ordinary simple annuity.
Interpret the result of a future-value calculation in context.

Prerequisite Bridge: Geometric Series and Compound Interest

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed number called the common ratio,

r

. For example,

3, 6, 12, 24, \ldots

is geometric with

r = 2

.

A geometric series is the sum of the terms of a geometric sequence. If the first term is

a

, the common ratio is

r

(where

r \neq 1

), and there are

n

terms, the sum is given by the formula

S_n = \frac{a(r^n - 1)}{r - 1}

.

You also need to recall compound interest. When a principal

P

earns an interest rate

i

per period, after

k

periods it grows to

P(1 + i)^k

. Each equal payment in an annuity will grow by this rule, which is why geometric sequences appear naturally.

S_n = \frac{a(r^n - 1)}{r - 1}

A geometric series sums terms that each share the same multiplicative ratio.
The sum formula is $S_n = \frac{a(r^n - 1)}{r - 1}$ for $r \neq 1$ .
Compound interest grows a single amount by a factor of $(1 + i)^k$ over $k$ periods.

What Is an Ordinary Simple Annuity?

An annuity is a sequence of equal, regular payments. The word ordinary means every payment is made at the end of each period — not the beginning. The word simple means the compounding period and the payment period are the same length. For example, monthly deposits with monthly compounding is a simple annuity; monthly deposits with daily compounding is not.

Three numbers fully describe an ordinary simple annuity: the regular payment amount

R

, the interest rate per period

i

, and the total number of payments

n

. When

i

is given as an annual rate, divide by the number of periods per year to find the rate per period.

The future value

FV

of the annuity is the total amount accumulated immediately after the last payment is made. This is the quantity we want to calculate, and it is the sum that connects annuities to geometric series.

Ordinary: payments occur at the end of each period.
Simple: the compounding period equals the payment period.
Defined by payment $R$ , rate per period $i$ , and number of payments $n$ .
Future value $FV$ is the total balance right after the final payment.

Building the Geometric Series from Individual Payments

Imagine

n

payments of amount

R

made at the end of periods

1, 2, 3, \ldots, n

. Each payment earns compound interest for a different number of periods. The first payment (made at end of period 1) earns interest for

n - 1

periods, so by the end of period

n

it has grown to

R(1+i)^{n-1}

. The second payment earns interest for

n - 2

periods and grows to

R(1+i)^{n-2}

. This pattern continues until the last payment, which is made right at the end and earns no interest, so its value is simply

R = R(1+i)^0

.

Writing all these values and adding them gives the future value as a sum:

FV = R(1+i)^{n-1} + R(1+i)^{n-2} + \cdots + R(1+i)^1 + R(1+i)^0

. Notice every term has the form

R(1+i)^k

for

k = 0, 1, 2, \ldots, n-1

.

This is exactly a geometric series. The first term (the smallest, written last above) is

a = R

, the common ratio is

r = (1+i)

, and the number of terms is

n

. Substituting into the geometric series sum formula gives the future-value formula for an ordinary simple annuity.

FV = R(1+i)^{n-1} + R(1+i)^{n-2} + \cdots + R(1+i)^0

The payment made at the end of period 1 earns interest for $n-1$ periods by the final date.
Each payment's grown value has the form $R(1+i)^k$ for a whole-number exponent $k$ .
The collection of all grown payment values forms a geometric series with $a = R$ and $r = (1+i)$ .
Summing the geometric series yields the future value $FV$ .

The Future Value Formula

Applying the geometric series sum formula

S_n = \frac{a(r^n - 1)}{r - 1}

with

a = R

and

r = (1+i)

gives the future value of an ordinary simple annuity.

The result is

FV = R \cdot \frac{(1+i)^n - 1}{i}

. The denominator simplifies because

r - 1 = (1+i) - 1 = i

. This formula is not a new idea — it is just the geometric series sum formula written with annuity labels.

Every part of this formula has a concrete meaning:

R

is the equal payment,

(1+i)^n

is the compound-growth factor applied to the full term, and

i

in the denominator captures the interest rate per period. If

i = 0

, no interest is earned and the balance would simply equal

n \times R

, which also makes intuitive sense.

FV = R \cdot \frac{(1+i)^n - 1}{i}

The future value formula comes directly from the geometric series sum formula.
The denominator $i$ comes from $r - 1 = (1+i) - 1 = i$ in the geometric series formula.
Each symbol maps directly to a feature of the annuity: $R$ is the payment, $i$ is the rate per period, and $n$ is the number of payments.

Reading the Formula and Checking Reasonableness

Before using the formula, always identify the three inputs:

R

,

i

, and

n

. Convert an annual interest rate to a per-period rate by dividing by the number of periods in a year. For monthly payments with an annual rate of 6%, the monthly rate is

i = 0.06 \div 12 = 0.005

.

After calculating

FV

, check reasonableness. The future value must be larger than

n \times R

(the total of all payments with no interest) because interest adds to the balance. If your answer is smaller than

n \times R

, recheck your calculation.

Also double-check that

n

is the total number of payments, not the number of years. If deposits are monthly for 3 years, then

n = 36

, not

3

.

i = \frac{i_{\text{annual}}}{m}

Convert the annual rate to a per-period rate before substituting: divide by the number of periods per year.
The future value $FV$ must always exceed $n \times R$ ; if it does not, there is an error.
Count the total number of payments, not the number of years, when determining $n$ .

How Each Payment Grows: 4-Payment Mini-Example (R = CAD 150, i = 0.004)

Payment #	Made at end of period	Periods earning interest	Value at end of period 4	Geometric term
1	Period 1	3	$150(1.004)^3 \approx 151.80$	$R \cdot r^{3}$
2	Period 2	2	$150(1.004)^2 \approx 151.20$	$R \cdot r^{2}$
3	Period 3	1	$150(1.004)^1 \approx 150.60$	$R \cdot r^{1}$
4	Period 4	0	$150(1.004)^0 = 150.00$	$R \cdot r^{0}$

Worked example

Savings Plan: Monthly Deposits Over Two Years

Ava deposits CAD 150 at the end of every month into an account that earns 4.8% per year compounded monthly. She makes no withdrawals. What is the future value of her annuity after 2 years?

Identify the annuity type
Payments are made monthly and interest compounds monthly, so the compounding period equals the payment period. Payments are at the end of each month. This is an ordinary simple annuity.
List the known values
The regular payment is $R = 150$ . The annual rate is 4.8%, so the monthly rate is $i = 0.048 \div 12 = 0.004$ . The number of monthly payments over 2 years is $n = 2 \times 12 = 24$ .
$R = 150, \quad i = 0.004, \quad n = 24$
Recognize the geometric series structure
Each payment grows with compound interest at rate $i = 0.004$ per month. The first payment grows for 23 months, the second for 22 months, and so on. The future values of all 24 payments form a geometric series with first term $a = R = 150$ and common ratio $r = 1 + 0.004 = 1.004$ .
$a = 150, \quad r = 1.004, \quad n = 24$
Write the future value using the geometric series sum formula
Substitute $a = 150$ , $r = 1.004$ , and $n = 24$ into $S_n = \frac{a(r^n - 1)}{r - 1}$ . Because $r - 1 = 0.004 = i$ , this is the same as writing $FV = \frac{150(1.004^{24} - 1)}{0.004}$ .
$FV = \frac{150(1.004^{24} - 1)}{0.004}$
Calculate the power
Use a calculator to evaluate $1.004^{24}$ .
$1.004^{24} \approx 1.10040$
Evaluate the numerator
Subtract 1 from the power, then multiply by 150: $150 \times (1.10040 - 1) = 150 \times 0.10040 \approx 15.060$ .
$150 \times 0.10040 = 15.060$
Divide by the denominator
The denominator is $r - 1 = 0.004$ . Divide the numerator by the denominator to get the future value.
$FV = \frac{15.060}{0.004} \approx 3765.00$
State the answer in context
Ava's account will hold approximately CAD 3765.00 at the end of 2 years.
$FV \approx 3765.00$

Answer: The future value is approximately CAD 3765.00.

Check: Total payments without interest:

24 \times 150 = \text{CAD } 3600.00

. Since CAD 3765.00 > CAD 3600.00, interest has been added correctly. The difference, approximately CAD 165.00, is the total interest earned, which is reasonable for a 4.8% annual rate over 2 years.

Worked example

RESP Contributions: Quarterly Deposits Over Five Years

Marcus contributes CAD 500 at the end of every quarter (every 3 months) into an education savings account earning 5.2% per year compounded quarterly. How much will be in the account right after his last deposit, 5 years from now? Also, how much of that total is interest?

Confirm the annuity type
Payments and compounding are both quarterly, and payments are at the end of each quarter. This is an ordinary simple annuity.
List the known values
The regular payment is $R = 500$ . The annual rate is 5.2%, so the quarterly rate is $i = 0.052 \div 4 = 0.013$ . Over 5 years there are $n = 5 \times 4 = 20$ quarterly payments.
$R = 500, \quad i = 0.013, \quad n = 20$
Connect to a geometric series
The 20 grown payment values form a geometric series with first term $a = 500$ and common ratio $r = 1 + 0.013 = 1.013$ , summed over $n = 20$ terms.
$a = 500, \quad r = 1.013, \quad n = 20$
Write the future value using the geometric series sum formula
Substitute $a = 500$ , $r = 1.013$ , and $n = 20$ into $S_n = \frac{a(r^n - 1)}{r - 1}$ . Since $r - 1 = 0.013 = i$ , this gives $FV = \frac{500(1.013^{20} - 1)}{0.013}$ .
$FV = \frac{500(1.013^{20} - 1)}{0.013}$
Calculate the power
Use a calculator to evaluate $1.013^{20}$ .
$1.013^{20} \approx 1.29220$
Evaluate the numerator
Subtract 1 and multiply by 500: $500 \times (1.29220 - 1) = 500 \times 0.29220 = 146.10$ .
$500 \times 0.29220 = 146.10$
Divide by the denominator
The denominator is $r - 1 = 0.013$ . Divide the numerator by the denominator to get the future value.
$FV = \frac{146.10}{0.013} \approx 11238.46$
Find the interest earned
Total contributions without interest equal $20 \times 500 = 10000$ . Subtract this from the future value to find total interest earned: $11238.46 - 10000 = 1238.46$ .
$11238.46 - 10000 = 1238.46$
State the answer in context
Marcus will have approximately CAD 11238.46 in the account after 5 years. Of that total, approximately CAD 1238.46 is interest earned.
$FV \approx 11238.46$

Answer: The future value is approximately CAD 11238.46, of which approximately CAD 1238.46 is interest.

Check: Total payments:

20 \times 500 = \text{CAD } 10000

. Since CAD 11238.46 > CAD 10000, the result is reasonable. The interest fraction is about 12.4% of total contributions over 5 years at 5.2% per year — a sensible ballpark given that most deposits earn interest for less than the full 5 years.

Common mistakes and how to avoid them

Using the annual interest rate directly instead of converting to the per-period rate. For example, using

i = 0.048

for monthly compounding instead of

i = 0.004

.

Correction: Always divide the annual rate by the number of compounding periods per year before substituting into the formula:

i = i_{\text{annual}} \div m

.

Setting

n

equal to the number of years instead of the total number of payments. For 3 years of monthly payments, a student writes

n = 3

instead of

n = 36

.

Correction: Multiply the number of years by the number of payments per year to get

n

. Monthly for 3 years gives

n = 3 \times 12 = 36

.

Confusing the geometric series first term. Some students set

a = R(1+i)^{n-1}

(the largest term) instead of

a = R

(the smallest term, which is the last payment).

Correction: The formula

FV = R \cdot \frac{(1+i)^n - 1}{i}

is derived with

a = R

and

r = (1+i)

. The last payment earns no interest and is the smallest term, equal to

R

.

Forgetting that the future value must be greater than the sum of all payments. A student accepts an answer of CAD 3200 for 24 payments of CAD 150 (total CAD 3600).

Correction: Interest always adds to the balance. If

FV < n \times R

, the calculation contains an error — recheck the rate and exponent.

Applying this formula to an annuity due (payments at the start of each period) without adjustment, or to a general annuity where the compounding and payment periods differ.

Correction: The formula

FV = R \cdot \frac{(1+i)^n - 1}{i}

applies only to ordinary simple annuities: end-of-period payments with matching compounding periods.

Lesson summary

An ordinary simple annuity has equal payments at the end of each period, with compounding occurring at the same frequency as payments.
Each payment grows with compound interest for a different number of periods, producing terms of the form $R(1+i)^k$ .
These terms form a geometric series with first term $a = R$ , common ratio $r = (1+i)$ , and $n$ terms.
Applying the geometric series sum formula $S_n = \frac{a(r^n-1)}{r-1}$ gives the future value formula $FV = R \cdot \frac{(1+i)^n - 1}{i}$ .
Before substituting, convert the annual interest rate to a per-period rate and count payments (not years) for $n$ .
A quick reasonableness check: the future value must always exceed the total of all payments $n \times R$ .

Check your understanding

Question 1

The future value of an ordinary simple annuity is found by applying which mathematical tool?

The sum of an arithmetic series
The sum of a geometric series
The product of a geometric sequence
Simple interest on the total payments

Show answer and explanation

The sum of a geometric series

Each payment grows by compound interest for a different number of periods, creating terms with a constant ratio

(1+i)

. Their total is a geometric series sum, not an arithmetic one.

Question 2

Jenna deposits CAD 400 at the end of every 6 months into an account earning 6% per year compounded semi-annually. What are the correct values of

R

,

i

, and

n

for 3 years?

$R = 400,\; i = 0.06,\; n = 3$
$R = 400,\; i = 0.03,\; n = 6$
$R = 400,\; i = 0.03,\; n = 3$
$R = 400,\; i = 0.06,\; n = 6$

Show answer and explanation

R = 400,\; i = 0.03,\; n = 6

The semi-annual rate is

i = 0.06 \div 2 = 0.03

. Over 3 years with 2 payments per year, the number of payments is

n = 3 \times 2 = 6

. So

R = 400

,

i = 0.03

,

n = 6

.

Question 3

In the geometric series that represents an ordinary simple annuity, what is the common ratio

r

?

$r = i$
$r = 1 - i$
$r = (1 + i)^n$
$r = 1 + i$

Show answer and explanation

r = 1 + i

Each successive payment earns one extra period of compound interest, so consecutive terms have a ratio of

(1 + i)

, which is the growth factor for one period.

Question 4

Carlos calculates the future value of his annuity and gets CAD 4800. His total payments add up to CAD 5100. What does this tell you?

The interest rate must be negative, so the answer is correct.
There is an error in the calculation because future value must exceed total payments.
The future value is correct because interest reduces the balance.
The answer is correct only if the interest rate is zero.

Show answer and explanation

There is an error in the calculation because future value must exceed total payments.

Interest always increases the account balance, so the future value of an ordinary simple annuity must be strictly greater than

n \times R

. A result smaller than the total payments signals a calculation error.

Key terms

Annuity: A sequence of equal payments made at regular intervals over a set period of time.
Ordinary annuity: An annuity in which each payment is made at the end of the payment period.
Simple annuity: An annuity in which the compounding period and the payment period are the same length of time.
Future value (FV): The total amount in an account immediately after the last payment of an annuity, including all interest earned.
Geometric series: The sum of the terms of a geometric sequence; each term is found by multiplying the previous term by the same constant ratio.
Common ratio (r): The constant multiplier between consecutive terms in a geometric sequence or series.
Interest rate per period (i): The annual interest rate divided by the number of compounding periods per year; the rate applied in each single period.
Number of payments (n): The total count of equal payments made over the life of the annuity.

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