Solve applications involving arithmetic and geometric sequences and series

Ontario Grade 11 Mathematics

Discrete Functions

Arithmetic and Geometric Applications — MCR3U Expectation C2.4

Many real situations follow a predictable numerical pattern. A salary that increases by the same dollar amount each year, or a population that doubles every decade, are both examples of sequences at work. In this lesson you will connect the formulas you already know for arithmetic and geometric sequences and series to practical problems. You will practise choosing the right formula, substituting carefully, and interpreting your answer in the language of the original question. Every new term is defined when it first appears, and every example is worked out step by step so you can follow the reasoning, not just the arithmetic.

What you will learn

Identify whether a real-world situation involves an arithmetic or geometric sequence or series.
Select and apply the correct formula to find a specific term or a partial sum in context.
Interpret the answer in terms of the original problem, including appropriate units.
Solve multi-step application problems that combine sequence and series reasoning.

Prerequisite Bridge: Sequences and Series at a Glance

A sequence is an ordered list of numbers. Each number in the list is called a term. The first term is written

t_1

, the second

t_2

, and so on. The term in position

n

is written

t_n

and is called the general term.

An arithmetic sequence has a constant difference between consecutive terms. That constant is called the common difference,

d

. The general term is

t_n = t_1 + (n-1)d

. The sum of the first

n

terms of an arithmetic sequence is

S_n = \frac{n}{2}(t_1 + t_n)

, or equivalently

S_n = \frac{n}{2}\bigl(2t_1 + (n-1)d\bigr)

.

A geometric sequence has a constant ratio between consecutive terms. That constant is called the common ratio,

r

. The general term is

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

. The sum of the first

n

terms of a geometric sequence is

S_n = \frac{t_1(r^n - 1)}{r - 1}

when

r \neq 1

.

A series is the sum of the terms of a sequence. When a problem asks for a total accumulated amount, you need a series formula. When it asks for a specific value at a particular time or position, you need a sequence formula.

Arithmetic: add or subtract the same amount each step — use $t_n = t_1 + (n-1)d$ .
Geometric: multiply by the same ratio each step — use $t_n = t_1 \cdot r^{n-1}$ .
Series formulas give totals; sequence formulas give individual terms.
Always identify $t_1$ , $d$ or $r$ , and $n$ before substituting into any formula.

Recognising the Pattern in an Application Problem

The hardest part of an application problem is deciding which type of sequence applies. Look at how the quantity changes from one step to the next. If the change is always the same fixed amount (add or subtract), the situation is arithmetic. If the change is always the same fixed multiplier (multiply or divide), the situation is geometric.

Read the problem once to understand the context, then extract three pieces of information: the starting value, how the value changes at each step, and what the question is actually asking — a specific term, or a total?

Common arithmetic clues in problem language: 'increases by,' 'decreases by,' 'earns an extra,' 'saves an additional,' or any phrase suggesting a fixed dollar or unit change per period.

Common geometric clues: 'doubles,' 'triples,' 'grows by a percentage,' 'depreciates by a percentage,' or 'halves.' A percentage change means you multiply by a fixed ratio each period — for example, growth of 20% per year means

r = 1.20

, and depreciation of 15% per year means

r = 0.85

.

Once you have identified the type, write down

t_1

,

d

or

r

, and

n

explicitly before touching any formula. This habit prevents the most common errors.

Fixed additive change → arithmetic sequence or series.
Fixed multiplicative change (including percentage) → geometric sequence or series.
'Specific term' questions → use $t_n$ ; 'total' questions → use $S_n$ .
Write out $t_1$ , $d$ or $r$ , and $n$ before substituting.
After solving, re-read the question to make sure your answer has the right units and meaning.

Arithmetic Applications: Steady Change Over Time

Suppose someone saves a fixed amount more each month than the previous month. The amounts saved each month form an arithmetic sequence because the increase is constant. If you want to know the total saved after several months, you need the arithmetic series formula.

Consider a scenario: a theatre has 20 seats in the first row, 23 seats in the second row, 26 in the third, and so on. The common difference is

d = 3

. To find the number of seats in the 15th row, use

t_n = t_1 + (n-1)d

, giving

t_{15} = 20 + (14)(3) = 62

seats. To find the total seats in the first 15 rows, use

S_{15} = \frac{15}{2}(t_1 + t_{15}) = \frac{15}{2}(20 + 62) = \frac{15}{2}(82) = 615

seats.

Notice that answering 'how many seats in row 15?' required the sequence formula, while 'how many seats in total across 15 rows?' required the series formula. The context drives the formula choice.

In salary and savings problems, check carefully whether the first given value is for period 1 or period 0. This affects the value of

n

you use, and getting it wrong by one is the single most common application error.

S_n = \frac{n}{2}(t_1 + t_n)

Arithmetic series formula: $S_n = \frac{n}{2}(t_1 + t_n)$ .
Find $t_n$ first when you only know $t_1$ and $d$ , then substitute into the series formula.
Double-check whether the first listed value corresponds to $n = 1$ or $n = 0$ .
Always state units in your final answer.

Geometric Applications: Proportional Change Over Time

When a quantity grows or shrinks by the same percentage each period, the amounts form a geometric sequence. The key is converting the percentage change into the common ratio

r

. If a quantity increases by 8% each year, then each year's value is 108% of the previous year's, so

r = 1.08

. If it decreases by 8%, then

r = 0.92

.

Geometric series arise when you want to accumulate totals over several periods of percentage change — for example, the total amount of medication absorbed over many doses, or the total distance a bouncing ball travels.

When evaluating

r^n

, always compute the exponent before multiplying by

t_1

. Use the order of operations carefully:

t_1 \cdot r^n

means apply the exponent to

r

first, then multiply by

t_1

.

One practical trap: if a bank account starts at CAD 1000 and earns 5% annual interest, the amount after 1 year is

t_2 = 1000 \times 1.05

, not

t_1

. The initial deposit is

t_1 = 1000

, year 1 balance is

t_2

, year 2 balance is

t_3

, and so on. So 'after 6 years' corresponds to

t_7

— unless the problem explicitly says year 1 is the first interest period, in which case 'after 6 years' is

t_6

. Read carefully and be consistent.

S_n = \frac{t_1(r^n - 1)}{r - 1}

Percentage increase of $p$ %: $r = 1 + \frac{p}{100}$ .
Percentage decrease of $p$ %: $r = 1 - \frac{p}{100}$ .
Geometric series formula: $S_n = \frac{t_1(r^n - 1)}{r - 1}$ , valid when $r \neq 1$ .
Exponentiate $r$ before multiplying by $t_1$ — respect order of operations.

Choosing Your Strategy and Checking Your Answer

Before you write a single formula, answer two questions: (1) Is the change additive or multiplicative? (2) Does the question ask for one specific term or a running total? Your answers to these two questions determine everything.

After calculating, always perform a reasonableness check. For an arithmetic series, the answer should be between

n \times t_1

and

n \times t_n

— the total lies between using only the first term and using only the last term for every period. For a geometric series with

r > 1

, the sum should be larger than

n \times t_1

and should grow quickly; if your sum is smaller than

t_1

, you made an error.

Also check units and rounding. Money problems in a Canadian context are typically rounded to the nearest cent. Counting problems (seats, people, items) must give whole-number answers — if you get a decimal, recheck your setup.

If a problem gives you

S_n

or

t_n

and asks you to find

n

, set up the formula, isolate the expression containing

n

, and solve. At the MCR3U level, these problems are structured so that

n

comes out as a whole number through careful algebra, allowing you to solve without any advanced techniques beyond rearranging the equation and checking whole-number candidates.

Two diagnostic questions: additive or multiplicative? One term or a total?
Reasonableness check: does the magnitude of your answer make sense in context?
Counting answers must be whole numbers; round money to cents.
When solving for $n$ , use algebra to isolate the term containing $n$ ; the answer will be a whole number at this level.

Choosing the Right Formula for an Application Problem

Situation type	Change per step	Question asks for	Formula to use
Arithmetic	Add $d$ each step	Value at step $n$	$t_n = t_1 + (n-1)d$
Arithmetic	Add $d$ each step	Total of $n$ terms	$S_n = \frac{n}{2}(t_1 + t_n)$
Geometric	Multiply by $r$ each step	Value at step $n$	$t_n = t_1 \cdot r^{n-1}$
Geometric	Multiply by $r$ each step	Total of $n$ terms	$S_n = \frac{t_1(r^n-1)}{r-1}$

Worked example

Arithmetic Series — Total Earnings Over a Contract

A graphic designer signs a 10-year contract. In year 1 she earns CAD 48 000. Each following year her salary increases by CAD 2 400. How much does she earn in total over the entire 10-year contract?

Identify the sequence type and extract known values
The salary increases by the same fixed amount each year, so this is an arithmetic sequence. The first term is $t_1 = 48\,000$ , the common difference is $d = 2\,400$ , and the contract covers $n = 10$ years. The question asks for the total earnings — a sum — so we need the series formula.
Find the salary in year 10
Before using the series formula, find $t_{10}$ using the general term formula. Substitute $n = 10$ , $t_1 = 48\,000$ , and $d = 2\,400$ into $t_n = t_1 + (n-1)d$ .
$t_{10} = 48\,000 + (10-1)(2\,400)$
Calculate $t_{10}$
Evaluate $(10 - 1) = 9$ , then $9 \times 2\,400 = 21\,600$ , and finally add to the starting salary.
$t_{10} = 48\,000 + 21\,600 = 69\,600$
Apply the arithmetic series formula
The sum of an arithmetic series is $S_n = \frac{n}{2}(t_1 + t_n)$ . Substitute $n = 10$ , $t_1 = 48\,000$ , and $t_{10} = 69\,600$ .
$S_{10} = \frac{10}{2}(48\,000 + 69\,600)$
Calculate the total
Simplify $\frac{10}{2} = 5$ and $48\,000 + 69\,600 = 117\,600$ , then multiply.
$S_{10} = 5 \times 117\,600 = 588\,000$

Answer: The designer earns a total of CAD 588 000 over the 10-year contract.

Check: A quick reasonableness check: if she earned her first-year salary every year, the total would be

10 \times 48\,000 = 480\,000

. If she earned her last-year salary every year, the total would be

10 \times 69\,600 = 696\,000

. The answer CAD 588 000 is exactly halfway between these two extremes, which is correct for an arithmetic series — the average salary is

\frac{48\,000 + 69\,600}{2} = 58\,800

and

10 \times 58\,800 = 588\,000

. ✓

Worked example

Geometric Series — Total Mass of Recycled Materials

A recycling programme collects 400 kg of material in its first month. Each month after that, the amount collected is 1.15 times the amount collected the previous month (a 15% increase). What is the total mass of material collected over the first 8 months? Round your answer to the nearest kilogram.

Identify the sequence type and extract known values
Each month's collection is multiplied by the same factor, 1.15, so this is a geometric sequence. The first term is $t_1 = 400$ , the common ratio is $r = 1.15$ , and we want the total over $n = 8$ months. A total means we need the geometric series formula.
Write the geometric series formula and substitute
The sum of the first $n$ terms of a geometric series is $S_n = \frac{t_1(r^n - 1)}{r - 1}$ . Substitute $t_1 = 400$ , $r = 1.15$ , and $n = 8$ .
$S_8 = \frac{400(1.15^8 - 1)}{1.15 - 1}$
Evaluate the power $1.15^8$
Calculate $1.15^8$ step by step to keep accuracy. First, $1.15^2 = 1.3225$ . Then $1.15^4 = 1.3225^2 \approx 1.74900625$ . Finally, $1.15^8 = 1.74900625^2 \approx 3.05902321$ . Keep several decimal places to avoid rounding error in the middle of the calculation.
$1.15^8 \approx 3.05902321$
Simplify the numerator
Replace $1.15^8$ with the approximate value, evaluate $1.15^8 - 1$ , and multiply by 400.
$S_8 = \frac{400(3.05902321 - 1)}{0.15} = \frac{400 \times 2.05902321}{0.15}$
Complete the division
Multiply $400 \times 2.05902321 = 823.60928$ , then divide by $0.15$ .
$S_8 = \frac{823.60928}{0.15} \approx 5490.73$
Round and state the answer
The question asks for the nearest kilogram, so round $5490.73$ to $5491$ .
$S_8 \approx 5491 \text{ kg}$

Answer: The recycling programme collects approximately 5 491 kg of material over the first 8 months.

Check: A lower bound: if the programme collected 400 kg every month, the total would be

8 \times 400 = 3\,200

kg. Since the amounts are growing, the true total must be larger than 3 200 kg, and 5 491 kg is indeed larger. As a spot check,

t_8 = 400 \times 1.15^7 \approx 400 \times 2.66002 \approx 1064

kg. The average of

t_1 = 400

and

t_8 \approx 1064

is about 732, and

8 \times 732 \approx 5856

. For a geometric series the weighted average sits slightly below the simple midpoint of first and last, so 5 491 kg is a plausible and consistent result. ✓

Common mistakes and how to avoid them

Using the sequence formula when the question asks for a total, or using the series formula when the question asks for a specific term.

Correction: Ask yourself: does the question want one value at a particular step, or the accumulated total of many steps? One value →

t_n

. Total →

S_n

.

Converting a percentage change to a common ratio incorrectly — for example, writing

r = 15

for a 15% increase instead of

r = 1.15

.

Correction: A 15% increase means each term is 115% of the previous one, so

r = 1 + 0.15 = 1.15

. A 15% decrease gives

r = 1 - 0.15 = 0.85

.

Off-by-one error in

n

— for example, treating 'after 6 years' as

n = 6

when the problem's setup means year 6 is actually

t_7

.

Correction: Clearly define what

t_1

represents in the context of the problem, then count forward consistently. Writing out the first few terms in a short table often reveals the correct value of

n

.

Multiplying

t_1

by

r^n

instead of

r^{n-1}

when finding a specific term, giving an answer that is one ratio too large.

Correction: The general term formula is

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

, not

t_1 \cdot r^n

. The first term requires zero multiplications by

r

, so the exponent is

n - 1

.

Rounding an intermediate result such as

r^n

to too few decimal places, causing a large error in the final answer for geometric series.

Correction: Keep at least four or five decimal places in all intermediate steps. Only round the final answer to the precision the problem requests.

Lesson summary

An arithmetic sequence has a constant difference $d$ ; a geometric sequence has a constant ratio $r$ . Identifying which type applies is the first step in every application problem.
Use the general term formula ( $t_n$ ) when the problem asks for a value at a specific position, and the series formula ( $S_n$ ) when it asks for a cumulative total.
For arithmetic series: $S_n = \frac{n}{2}(t_1 + t_n)$ . Find $t_n$ first if it is not given directly.
For geometric series: $S_n = \frac{t_1(r^n - 1)}{r - 1}$ . Convert any percentage change to a decimal ratio before substituting.
Always define $t_1$ , $d$ or $r$ , and $n$ explicitly before using a formula, and perform a reasonableness check after calculating.
Off-by-one and percentage-conversion errors are the most common mistakes; writing out the first few terms of the sequence in context will catch both.

Check your understanding

Question 1

A landscaping company plants 30 trees in week 1 and plants 5 more trees each subsequent week. How many trees does it plant in week 9?

70 trees
75 trees
65 trees
360 trees

Show answer and explanation

70 trees

This is arithmetic with

t_1 = 30

,

d = 5

, and

n = 9

. Using

t_9 = 30 + (9-1)(5) = 30 + 40 = 70

. Option B uses

n = 9

instead of

n - 1 = 8

as the multiplier, giving

30 + 45 = 75

. Option C uses

d = 4

by mistake. Option D is the series sum

S_9

, not a single term.

Question 2

A ball is dropped and bounces to 60% of its previous height each time. The first bounce reaches 250 cm. Which expression gives the total height covered by the first 5 bounces (upward only)?

$250 \times 0.60^4$
$5 \times 250 \times 0.60$
$\frac{250(0.60^5 - 1)}{0.60 - 1}$
$\frac{250(1 - 0.60^5)}{1 - 0.60}$

Show answer and explanation

\frac{250(1 - 0.60^5)}{1 - 0.60}

The bounce heights form a geometric sequence with

t_1 = 250

and

r = 0.60

. The total of 5 terms is

S_5 = \frac{250(0.60^5 - 1)}{0.60 - 1}

. Since both numerator and denominator are negative, they cancel and the expression equals

\frac{250(1 - 0.60^5)}{1 - 0.60}

, which is option D. Option C is algebraically equivalent but written with two negatives that must be cancelled — option D is the cleaner standard form. Option A gives only the 5th term. Option B is not a valid sequence or series formula.

Question 3

An arithmetic series has

t_1 = 12

,

d = 4

, and

n = 20

. What is

S_{20}

?

480
1 000
760
88

Show answer and explanation

1 000

First find

t_{20} = 12 + (19)(4) = 12 + 76 = 88

. Then

S_{20} = \frac{20}{2}(12 + 88) = 10 \times 100 = 1\,000

. Option A is

20 \times 24 = 480

, using twice

t_1

instead of

t_1 + t_{20}

. Option C comes from only summing to

n = 10

by mistake. Option D is just

t_{20}

alone, not the series sum.

Question 4

A social media post is shared by 3 people on day 1. Each of those people shares it with 3 new people on day 2, and so on (each day's recipients each share with 3 new people). How many people in total have seen the post by the end of day 5, including the original 3?

243
363
360
3 906

Show answer and explanation

363

The daily new viewers form a geometric sequence:

t_1 = 3

and

r = 3

. The total after 5 days is

S_5 = \frac{3(3^5 - 1)}{3 - 1} = \frac{3(243 - 1)}{2} = \frac{3 \times 242}{2} = \frac{726}{2} = 363

. Option A (

243 = 3^5

) is only the number of new viewers on day 5. Option C (360) results from an arithmetic error when adding the five terms. Option D is far too large and comes from misapplying the formula with an incorrect ratio.

Key terms

Sequence: An ordered list of numbers where each number is called a term. The position of a term is given by its index, usually written as $n$ .
Series: The sum of the terms of a sequence. A partial sum $S_n$ is the sum of the first $n$ terms.
Arithmetic sequence: A sequence in which each term is obtained from the previous term by adding a fixed amount called the common difference $d$ .
Geometric sequence: A sequence in which each term is obtained from the previous term by multiplying by a fixed amount called the common ratio $r$ .
Common difference ( $d$ ): The constant amount added at each step in an arithmetic sequence. It can be positive (increasing), negative (decreasing), or zero (constant).
Common ratio ( $r$ ): The constant factor multiplied at each step in a geometric sequence. Values of $r$ between 0 and 1 produce decreasing sequences; values greater than 1 produce increasing sequences.
General term ( $t_n$ ): A formula that gives the value of any term in a sequence based on its position number $n$ .
Partial sum ( $S_n$ ): The result of adding together the first $n$ terms of a sequence. The formula for $S_n$ differs for arithmetic and geometric sequences.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation C2.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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