Introduction to Linear Polynomials · Lesson 3 of 7
Exploring Linear Patterns
“When numbers march in a pattern, algebra quietly learns their choreography.”
• Recognise a linear pattern from a sequence or visual arrangement. • Find a rule for the nth stage of a linear pattern. • Use constant difference to explain why a pattern is linear. • Model both increasing and decreasing situations with linear expressions. • Interpret piecewise-looking real situations such as fare rules carefully.
Linear Pattern
A linear pattern is a pattern in which the quantity changes by the same amount each time the stage number increases by one. This means that as we move from one stage to the next, the difference between consecutive values remains constant. The pattern may be shown using shapes, numbers, objects, or a table, but the important idea is the way the quantity changes.
So, when we study a linear pattern, we should not focus only on how the picture looks. We should look carefully at the values and ask: how much is being added or subtracted at each step? If this change stays the same throughout the pattern, then the relationship is linear. This constant change is what allows us to describe the pattern using a linear expression.
A linear pattern is a sequence in which the difference between consecutive terms is constant.
Growing Pattern of Square Tiles
Consider a pattern with 1 tile at Stage 1, 3 tiles at Stage 2, 5 tiles at Stage 3 and 7 tiles at Stage 4. Each new stage adds exactly 2 tiles.
| Stage n | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Number of tiles | 1 | 3 | 5 | 7 | 9 | 11 | 13 |
The constant difference is 2. To find a general rule, compare the stage number n with the number of tiles. Twice the stage number gives 2n, but the actual number is always one less. Therefore the rule is 2n − 1.
Problem
How many tiles are in Stage 15 and Stage 26? Which stages contain 21 and 47 tiles?
- 1.Stage 15: T(15) = 2(15) − 1 = 29.
- 2.Stage 26: T(26) = 52 − 1 = 51.
- 3.For 21 tiles: 2n − 1 = 21, so 2n = 22 and n = 11.
- 4.For 47 tiles: 2n − 1 = 47, so 2n = 48 and n = 24.
A Decreasing Linear Pattern
Linear patterns do not have to grow. Suppose Bela starts with ₹100 and spends ₹5 each day. The amount remaining decreases by the same amount every day.
| Day n | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Amount left ₹ | 100 | 95 | 90 | 85 | 80 |
Problem
Bela starts with ₹100 and spends ₹5 per day. After how many days will ₹40 remain?
- 1.Use A(n) = 100 − 5n.
- 2.Set 100 − 5n = 40.
- 3.Then 5n = 60.
- 4.So n = 12.
- 5.After 12 days, ₹40 remains.
A Fare Rule with an Initial Condition
An auto-rickshaw fare may stay fixed for an initial distance and then increase by a constant amount. Suppose the fare is ₹25 for the first 2 km and then increases by ₹15 for each additional kilometre. For a trip of n km, with n ≥ 2, the extra distance is n − 2 km.
| Distance km | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Fare ₹ | 25 | 25 | 40 | 55 | 70 | 85 |
Problem
For how many kilometres will the fare be ₹130?
- 1.Use F(n) = 15n − 5 for n ≥ 2.
- 2.Set 15n − 5 = 130.
- 3.Then 15n = 135.
- 4.So n = 9.
- 5.A fare of ₹130 corresponds to 9 km.
The expression 15n − 5 describes the fare rule only after the initial condition is accounted for. The context tells us that it applies for n ≥ 2.
Building Linear Expressions from Context
The same structure appears in savings, population, reading progress, area and volume. The general pattern is starting value plus or minus a fixed change multiplied by the number of steps.
Practice Problems
- A student has ₹500 and receives ₹150 each month. Write the amount after the nth month and find the amount after 6 months.
- A rally starts with 120 members and 9 members leave every hour. Write the number remaining after n hours.
- A rectangle has fixed length 13 cm. Write its area as a linear expression in breadth b and find the areas for b = 12, 10 and 8.
- A rectangular box has fixed length 7 cm and breadth 11 cm. Write volume as a linear expression in height h and evaluate it for h = 5, 9 and 13.
- A 500-page book is being read at 20 pages per day. Write the number of pages left after n days and find how many remain after 15 days.
Key Takeaways
• A linear pattern has a constant difference between successive terms. • The nth term of a linear pattern is a linear expression in n. • Linear patterns can increase or decrease. • Real rules may include a starting condition that must be interpreted carefully. • A common model is initial value plus or minus constant change × number of steps.
Next, we separate linear increase and linear decrease more clearly and study them as linear growth and linear decay.