C. Discrete Functions · Sequences and Series

Solving Problems with Arithmetic and Geometric Sequences

C2.4 C2.2 C2.1About 75 minutes

Learning goals

I can decide whether a real situation is modelled by an arithmetic or a geometric sequence.
I can choose a, d or r and n correctly, including when the situation starts at time 0.
I can use the general term to answer questions about a real situation, with a concluding sentence and units.
I can find the first term that passes a target value.

Words to know

model
A mathematical description (here, a sequence) of a real situation, used to make predictions.

constant change
The same amount added or subtracted each step. It leads to an arithmetic sequence.

constant percent change
Multiplying by the same factor each step, e.g. keeping 75% means r = 0.75 and growing by 6% means r = 1.06. It leads to a geometric sequence.

rebound ratio
The fraction of its previous height that a ball reaches on each bounce.

From words to sequences

Sequences are not just lists on a page. A stack of chairs, the seats in an arena, a bouncing ball, a growing colony of bacteria and a car that loses value each year can all be described by arithmetic or geometric sequences. The hard part of a word problem is rarely the formula. It is reading the situation carefully: what kind of change is happening, what is the first term, and which term does the question ask for?

A four-step plan

Type: the same amount each step → arithmetic. The same factor or percent each step → geometric.
First term and change: write a = …, and d = … or r = …. Watch percents: losing 15% means r = 0.85; gaining 6% means r = 1.06.
Which term? Decide what t₁ stands for. If the situation starts at "time 0", then the value after k steps is tₖ₊₁. A quick table of the first two or three terms prevents most mistakes.
Answer with a sentence and units, and check that the answer makes sense.

Worked example

Example 1: Stacking chairs (arithmetic)

One plastic chair is 80 cm tall. Each extra chair stacked on top adds 6 cm. (a) How tall is a stack of 15 chairs? (b) What is the largest number of chairs that can be stacked to fit under a doorway 2.1 m high?

The height grows by the same amount each chair: arithmetic, a = 80, d = 6. tₙ is the height of a stack of n chairs.
(a) t₁₅ = 80 + 14(6) = 80 + 84 = 164 cm.
(b) 2.1 m = 210 cm. Solve 80 + 6(n − 1) ≤ 210: 6(n − 1) ≤ 130, n − 1 ≤ 21.67, so n ≤ 22.67.
n must be a whole number, so 22 chairs (height 80 + 21(6) = 206 cm). 23 chairs would be 212 cm, too tall.

(a) 164 cm (b) 22 chairs

Worked example

Example 2: Arena seating (arithmetic)

A section of a hockey arena has 18 seats in the front row, and each row has 3 more seats than the row in front of it. Which row has 60 seats?

Arithmetic: a = 18, d = 3. Solve tₙ = 60.
18 + 3(n − 1) = 60 → 3(n − 1) = 42 → n − 1 = 14 → n = 15.
Row 15 has 60 seats.

Row 15

A ball dropped from 200 cm rebounds to 75% of its previous height each time: 150 cm, 112.5 cm, 84.375 cm, about 63.3 cm, and so on. The rebound heights form a geometric sequence. (Not to scale horizontally.)

Worked example

Example 3: The bouncing ball (geometric)

A ball is dropped from 200 cm. Each bounce reaches 75% of the previous height. (a) How high does it rise on the 5th bounce? (b) On which bounce does it first rise less than 10 cm?

Each height is 0.75 times the one before: geometric, r = 0.75.
Let t₁ be the height of the first bounce: t₁ = 0.75 × 200 = 150 cm. So tₙ = 150(0.75)ⁿ⁻¹ is the height of bounce n.
(a) t₅ = 150(0.75)⁴ = 150(0.3164…) ≈ 47.5 cm.
(b) Solve 150(0.75)ⁿ⁻¹ < 10, so 0.75ⁿ⁻¹ < 0.0667. Try powers: 0.75⁹ ≈ 0.0751 (not yet), 0.75¹⁰ ≈ 0.0563 (yes). So n − 1 = 10, n = 11.
Check: t₁₀ = 150(0.75)⁹ ≈ 11.3 cm and t₁₁ ≈ 8.4 cm. The 11th bounce is the first under 10 cm.

(a) about 47.5 cm (b) the 11th bounce

Worked example

Example 4: Bacteria (geometric, starting at time 0)

A culture starts with 500 bacteria, and the number doubles every 20 minutes. How many bacteria are there after 3 hours?

Doubling: geometric, r = 2.
3 hours = 180 minutes = 9 doubling periods.
Careful with the index: at time 0 there are 500 bacteria, so t₁ = 500. After 1 period it is t₂, …, after 9 periods it is t₁₀.
t₁₀ = 500(2)⁹ = 500(512) = 256 000 bacteria.

256 000 bacteria

When a situation starts at time 0, the starting amount is t₁. After 9 doubling periods you are at t₁₀, not t₉.

Worked example

Example 5: Value of a snowmobile (geometric)

A snowmobile costs $12 000 new and loses 15% of its value each year. (a) What is it worth after 4 years? (b) After how many full years is it first worth less than half of its original price?

Losing 15% means keeping 85% each year: geometric, r = 0.85.
Value after k years = 12 000(0.85)ᵏ (this is t_{k+1} if t₁ = 12 000 is the new price).
(a) 12 000(0.85)⁴ = 12 000(0.52200625) ≈ $6264.08.
(b) Need 0.85ᵏ < 0.5. 0.85⁴ ≈ 0.522 (not yet), 0.85⁵ ≈ 0.444 (yes). After 5 years it is worth about 12 000(0.4437) ≈ $5324.46, less than $6000.

(a) about $6264.08 (b) after 5 years

Just for fun

A bouncing ball that keeps 75% of its height each time is a geometric sequence.

A bouncing ball that keeps 100% of its height is either a perfect physics problem or a very suspicious magic trick.

Hands-on activity

Measure a bouncing ball

You need: a ball that bounces well (a tennis ball or rubber ball), a measuring tape taped to a wall, and a helper with a phone camera (slow-motion video helps).

Hold the bottom of the ball at 100 cm and let it drop (do not throw it). Film the bounces against the tape.
Play the video back and record the height of the first 3 or 4 rebounds (use the bottom of the ball each time).
Divide each height by the one before it. Are the ratios roughly constant? Find their average, r.
Write a general term for the height of the nth bounce. Predict bounce 6 and the first bounce under 5 cm.
Repeat with a different ball or a different floor (carpet versus hard floor). How does r change?

What to look for: a tennis ball on a hard floor often keeps somewhere around half its height each bounce; for example heights 55, 30, 17 cm give ratios about 0.55 and 0.57, so r ≈ 0.56 and tₙ ≈ 55(0.56)ⁿ⁻¹. Bounce 6 ≈ 55(0.56)⁵ ≈ 3.0 cm. Carpet gives a smaller r. Real balls are not perfectly geometric, but the model is close.

Your turn

A theatre has 22 seats in row 1, and each row has 2 more than the one before. How many seats are in row 18?

A ball is dropped from 3 m and rebounds to 60% of its previous height each time. How high is the 4th bounce?

A town of 8000 people grows by 3% per year. Estimate its population after 10 years.

A taxi charges $4.25 to start and $1.75 per kilometre. Is the cost after each whole kilometre arithmetic or geometric? Find the cost for 12 km.

A colony of 200 cells triples every hour. How many cells are there after 6 hours?

In the bouncing-ball problem from 3 m with r = 0.6, which bounce is the first lower than 10 cm?

More practice: warm-up

A plant is 12 cm tall and grows 1.5 cm per week. Arithmetic or geometric?

An investment grows by 5% each year. What is r?

A car loses 20% of its value each year. What is r?

A sequence models a situation that starts at time 0. If you want the value after 5 steps, which term do you need?

A ball rebounds to 2/3 of its previous height. A bounce is 90 cm high. How high is the next one?

More practice: on your own

A new bike path is built in stages: 2.5 km in the first year and 0.75 km more each year than the year before. How many km are built in year 9?

The light passing through one sheet of a tinted film is 85% of the light that hits it. What percent of the light gets through 6 sheets?

A pendulum swings 50 cm on its first swing, and each swing is 4% shorter than the one before. How long is the 10th swing?

The number of people who have heard a rumour doubles every day, starting with 3 people on day 1. On which day do more than 1000 people know?

A stairway's first step is 18 cm above the floor and each step rises 18 cm more. A landing is at 2.88 m. Which step is level with the landing?

Challenge

Two cities: Maplewood has 40 000 people and grows by 1200 people per year; Pine Ridge has 32 000 people and grows by 4% per year. After how many years does Pine Ridge first have more people?

A ball rebounds to the same fraction of its height each time. Its 2nd bounce is 72 cm and its 4th bounce is 40.5 cm. Find the rebound ratio and the height it was dropped from.

A geometric model for a town's population gives 5000 in 2020 and 5618 in 2024 (to the nearest person). Find the yearly growth rate and predict the population in 2030.

Check your understanding

Quiz

A ski club has 40 members and adds 5 new members each month. This is best modelled by:

A canoe loses 10% of its value each year. What is r?

A ball dropped from 2 m rebounds to 50% of its height each time. How high is the 3rd bounce?

A population of 1000 doubles every year. Which term gives the population after 4 years if t₁ = 1000 is the population now?

Row 1 of a section has 15 seats and each row has 4 more. Which row has 67 seats?

For tₙ = 150(0.8)ⁿ⁻¹, which is the first term less than 50?

For parents and teachers

Big idea: Expectation C2.4 asks students to solve problems involving arithmetic and geometric sequences, including real-world ones. The skills are reading the situation (constant amount or constant percent?), choosing a, d or r, and getting the term number right.

How to help: Ask your teen to make a quick table of the first two or three values before using any formula. Ask "What does t₁ stand for here?" and "Does the answer make sense?"

Common difficulties:

Off-by-one errors when the situation starts at time 0 (Example 4).
Writing r = 0.15 for "loses 15%" instead of r = 0.85.
Treating a rebound height as the first term when the drop height is given (Example 3: the first bounce is 75% of the drop).
Rounding too early. Keep full calculator values and round at the end.

Compound interest is studied separately in the Financial Applications unit.