To translate a story, decide whether it adds the same amount or multiplies by the same factor each step. Then name a, d or r, and n, and use the formula for that type.
This lesson is part of progressions. It uses the skills from the lessons on arithmetic and geometric progressions.
How do I decide which progression a story is?
Look for the word that describes the change. “More” followed by an amount means add. “Percent” or “times” means multiply.
| Words in the story | Pattern | Type |
|---|---|---|
| “RM20 more each month” | Add 20 | Arithmetic, d = 20 |
| “2 more seats in each row” | Add 2 | Arithmetic, d = 2 |
| “grows by 10% each week” | Multiply by 1.1 | Geometric, r = 1.1 |
| “halves each time” | Multiply by 0.5 | Geometric, r = 0.5 |
A percentage change is always geometric, because the amount added depends on the current size.
Worked example 1: savings
Here is an original story. A student saves RM100 in the first month and RM20 more each month than the month before. How much does she save in month 12, and how much in total over 12 months?
The amounts add 20 each time, so this is arithmetic with a = 100, d = 20, n = 12.
T₁₂ = 100 + 11 × 20 = RM320.
S₁₂ = 12/2 × (100 + 320) = 6 × 420 = RM2 520.
Worked example 2: rows of seats
A hall has 14 seats in the first row, and each row has 2 more seats than the row in front. There are 20 rows. Find the seats in the last row and in the whole hall.
This is arithmetic with a = 14, d = 2, n = 20.
T₂₀ = 14 + 19 × 2 = 14 + 38 = 52 seats.
S₂₀ = 20/2 × (14 + 52) = 10 × 66 = 660 seats.
Worked example 3: growth by a percentage
A fictional plant is 50 cm tall now and grows by 10% each week. Find its height after 5 weeks.
A 10% rise multiplies by 1.1 each week, so this is geometric with r = 1.1. The height now is the first term, T₁. After 5 weeks we want the sixth term, T₆.
T₆ = 50 × 1.1⁵ = 50 × 1.61051 = 80.5255, so the height is 80.53 cm.
The wording matters here. “Now” is T₁, “after 1 week” is T₂, and so “after 5 weeks” is T₆. Writing T₅ would be one term short.
The mistake that costs marks
The common slip is to treat “grows by 10%” as “adds 10”. For the plant that gives 50 + 5 × 10 = 100 cm, which is far too large and does not use the percentage at all.
| Step | Wrong | Right |
|---|---|---|
| Reading “10%” | Add 10 each week | Multiply by 1.1 each week |
| Type | Arithmetic | Geometric |
| Height after 5 weeks | 100 cm | 80.53 cm |
Another way to catch this is to test the first step. A 10% rise on 50 cm is 5 cm, not 10 cm, so the second term is 55.
Check yourself
A runner trains for 20 minutes on day 1 and adds 5 minutes each day. Find the training time on day 10 and the total time over the first 10 days.
Answer
The same amount is added each day, so this is arithmetic with a = 20, d = 5, n = 10.
T₁₀ = 20 + 9 × 5 = 65 minutes.
S₁₀ = 10/2 × (20 + 65) = 5 × 85 = 425 minutes.
Check with the other formula: 5 × [40 + 9 × 5] = 5 × 85 = 425.
What to study next
Mixed questions help you practise the decision. Try the progressions practice set, and use the word-problem structure worksheet to set out the facts in a story before you start.
If you want a teacher to give you fresh stories and pause you at the decision, see online one-to-one Additional Mathematics tuition.