Arithmetic and Geometric Sequences and Series Compared
When a word problem describes something growing over time, two questions help you choose the right formula: Does it increase by a fixed amount or a fixed percentage? And do you need one term or a total?
For Winnipeg students reviewing sequences and series, the comparison below shows how these decisions work in a single example.
Fixed Amount or Fixed Percentage
An arithmetic sequence increases or decreases by the same amount each step. This amount is called the common difference, d.
For example:
12, 15, 18, 21, …
Each term increases by 3, so d = 3.
A geometric sequence multiplies by the same factor each step. This factor is called the common ratio, r.
For example:
12, 14.4, 17.28, 20.736, …
Each term increases by 20%. Keeping 100% and adding 20% gives a multiplier of 1.20, so r = 1.20.
A fixed percentage increase produces a geometric sequence because the multiplier stays constant, even though the amount added changes.
One Term or a Running Total
A sequence lists individual terms. A series adds those terms together.
Compare these questions:
How much is collected on day 5? Find the fifth term, a₅.
How much is collected in total over the first 5 days? Find the sum, S₅.
The word total is an important clue: you need to add the daily amounts, rather than find just one day’s amount.
Comparing Two Fundraising Plans
A school club compares two models for its daily fundraising. Both start with $12 collected on day 1.
Plan A: The amount collected increases by $3 each day.
Plan B: The amount collected increases by 20% each day.
Assume both patterns continue exactly as described. Keep full precision during calculations and round the displayed amounts to the nearest cent.
We will compare which plan first reaches:
At least $40 collected in a single day.
At least $90 collected in total.
Part 1: Which Plan First Collects at Least $40 in One Day?
This asks for an individual day’s amount, so use term formulas.
For an arithmetic sequence:
aₙ = a₁ + (n − 1)d
For a geometric sequence:
aₙ = a₁rⁿ⁻¹
Here, a₁ is the first term and n is the day number. The exponent or multiplier uses n − 1 because there are n − 1 changes between day 1 and day n.
Plan A
Substitute a₁ = 12 and d = 3:
aₙ = 12 + 3(n − 1)
Find when the amount reaches at least $40:
12 + 3(n − 1) ≥ 40
Subtract 12:
3(n − 1) ≥ 28
Divide by 3, then add 1:
n − 1 ≥ 28/3
n ≥ 10⅓
The day number must be a whole number, so the first possible day is day 11.
Check:
Day 10: 12 + 3(9) = $39
Day 11: 12 + 3(10) = $42
Plan B
Substitute a₁ = 12 and r = 1.20:
aₙ = 12(1.20)ⁿ⁻¹
Use a calculator table or evaluate successive whole-number days to find where the amount first reaches $40.
The values around the target are:
Day 7: 12(1.20)⁶ ≈ $35.83
Day 8: 12(1.20)⁷ ≈ $43.00
Since the amounts increase each day, these values confirm that Plan B first reaches the target on day 8.
Plan B reaches $40 in a single day first: day 8 compared with day 11 for Plan A.
Part 2: Which Plan First Collects at Least $90 in Total?
Now we need the sum of all daily amounts, so use series formulas.
For an arithmetic series:
Sₙ = n[2a₁ + (n − 1)d] ÷ 2
For a geometric series:
Sₙ = a₁(rⁿ − 1) ÷ (r − 1)
The geometric formula applies when r ≠ 1.
Plan A
Substitute a₁ = 12 and d = 3:
Sₙ = n[24 + 3(n − 1)] ÷ 2
Evaluate the running totals around $90:
First 4 days: 4[24 + 3(3)] ÷ 2 = $66
First 5 days: 5[24 + 3(4)] ÷ 2 = $90
Plan A first reaches the total on day 5.
Plan B
Substitute a₁ = 12 and r = 1.20:
Sₙ = 12(1.20ⁿ − 1) ÷ 0.20
Evaluate the running totals around $90:
First 5 days: 12(1.20⁵ − 1) ÷ 0.20 ≈ $89.30
First 6 days: 12(1.20⁶ − 1) ÷ 0.20 ≈ $119.16
Plan B first reaches the total on day 6.
Plan A reaches $90 in total first: day 5 compared with day 6 for Plan B.
Why the Results Are Different
Plan A adds more money during the earliest days, giving it a small lead in the running total. Plan B’s percentage increases become larger as its daily amount grows, allowing it to reach the higher single-day target sooner.
A plan can reach a daily target first without reaching a particular cumulative target first. Always check whether the question asks for a term or a sum.
For “first reaches” questions, find the first whole-number day when the amount equals or exceeds the target. Check the preceding day to confirm that it was still below the target.
Math Tutoring for Winnipeg Students
If choosing between sequence and series formulas feels confusing, individual tutoring can help you connect the wording of a question to the correct method.
At Tutor Advance, I offer online math tutoring for Winnipeg students, with worked examples and practice tailored to each student’s course. As your math tutor, I can help you build confidence with sequences and series and apply these ideas independently.
Contact Tutor Advance to discuss your math course, learning goals, and current tutoring availability.