Understanding Linear Sequences
What Makes a Sequence Linear?
A linear sequence is a list of numbers where the difference between one term and the next is always the same. This constant difference is called the common difference and is usually represented by the letter $d$. Think of it like climbing a staircase where each step is exactly the same height - that consistent step height is your common difference.
For example, consider the sequence: 2, 5, 8, 11, 14, ...
Let's check the differences:
- $5 - 2 = 3$
- $8 - 5 = 3$
- $11 - 8 = 3$
- $14 - 11 = 3$
Since the difference is always 3, this is a linear sequence with common difference $d = 3$.
Finding the nth Term Formula
The most powerful tool for working with linear sequences is the nth term formula. This formula lets you calculate any term in the sequence without listing all the previous terms. For a linear sequence, the nth term formula always follows this pattern:
Where:
$a_n$ = the nth term you want to find
$a_1$ = the first term in the sequence
$n$ = the position of the term (1st, 2nd, 3rd, etc.)
$d$ = the common difference
Let's find the nth term for our example sequence: 2, 5, 8, 11, 14, ...
Step 1: Identify $a_1 = 2$ and $d = 3$
Step 2: Substitute into the formula: $a_n = 2 + (n - 1) \times 3$
Step 3: Simplify: $a_n = 2 + 3n - 3 = 3n - 1$
So the nth term formula is $a_n = 3n - 1$
Let's verify: For the 4th term ($n = 4$), $a_4 = 3 \times 4 - 1 = 12 - 1 = 11$ ✓
Types of Linear Sequences
Linear sequences can appear in different forms, but they all share the same fundamental property of having a constant difference between terms.
| Type | Description | Example | Common Difference |
|---|---|---|---|
| Increasing | Terms get larger as you move through the sequence | 5, 8, 11, 14, 17, ... | $d = 3$ |
| Decreasing | Terms get smaller as you move through the sequence | 20, 17, 14, 11, 8, ... | $d = -3$ |
| Constant | All terms are exactly the same | 7, 7, 7, 7, 7, ... | $d = 0$ |
| Fractional | Common difference is a fraction or decimal | 1, 1.5, 2, 2.5, 3, ... | $d = 0.5$ |
Linear Sequences in the Real World
Linear sequences appear everywhere in our daily lives. Recognizing these patterns helps us make predictions and solve practical problems.
Example 1: Saving Money
Suppose you decide to save $10 each week. Your savings would form a linear sequence:
Week 1: $10, Week 2: $20, Week 3: $30, Week 4: $40, ...
Common difference: $d = 10$
nth term formula: $a_n = 10 + (n - 1) \times 10 = 10n$
How much will you have after 15 weeks? $a_{15} = 10 \times 15 = 150$ dollars
Example 2: Plant Growth
A sunflower grows 3 cm each day. If it starts at 15 cm tall:
Day 1: 15 cm, Day 2: 18 cm, Day 3: 21 cm, Day 4: 24 cm, ...
Common difference: $d = 3$
nth term formula: $a_n = 15 + (n - 1) \times 3 = 12 + 3n$
How tall will it be after 3 weeks (21 days)? $a_{21} = 12 + 3 \times 21 = 75$ cm
Example 3: Temperature Change
On a cold day, the temperature drops by 2°C each hour starting from 10°C:
Hour 1: 10°C, Hour 2: 8°C, Hour 3: 6°C, Hour 4: 4°C, ...
Common difference: $d = -2$
nth term formula: $a_n = 10 + (n - 1) \times (-2) = 12 - 2n$
Graphing Linear Sequences
When you plot the terms of a linear sequence on a graph, you'll always get points that lie on a straight line. This is why they're called "linear" sequences! The position number ($n$) goes on the x-axis, and the term value ($a_n$) goes on the y-axis.
For the sequence $a_n = 3n - 1$:
- When $n = 1$, $a_n = 2$ → point (1, 2)
- When $n = 2$, $a_n = 5$ → point (2, 5)
- When $n = 3$, $a_n = 8$ → point (3, 8)
- When $n = 4$, $a_n = 11$ → point (4, 11)
If you plot these points, you'll see they form a perfect straight line. The slope of this line equals the common difference $d = 3$.
Advanced Concepts: Arithmetic Progressions[1]
In higher mathematics, linear sequences are formally called arithmetic sequences or arithmetic progressions (AP). The study of these sequences includes finding the sum of multiple terms, which has its own useful formula.
Where $S_n$ is the sum of the first $n$ terms.
Let's find the sum of the first 5 terms of our sequence: 2, 5, 8, 11, 14
Using the first formula: $S_5 = \frac{5}{2} \times [2 \times 2 + (5 - 1) \times 3] = \frac{5}{2} \times [4 + 12] = \frac{5}{2} \times 16 = 40$
Using the second formula: $S_5 = \frac{5}{2} \times (2 + 14) = \frac{5}{2} \times 16 = 40$
Check by adding: $2 + 5 + 8 + 11 + 14 = 40$ ✓
Common Mistakes and Important Questions
Q: Is the sequence 1, 4, 9, 16, 25,... a linear sequence?
No, this is not a linear sequence. Let's check the differences:
- $4 - 1 = 3$
- $9 - 4 = 5$
- $16 - 9 = 7$
- $25 - 16 = 9$
The differences are 3, 5, 7, 9, which are not constant. This is actually the sequence of perfect squares: $1^2, 2^2, 3^2, 4^2, 5^2,...$ which is a quadratic sequence, not a linear one.
Q: Can a linear sequence have a negative common difference?
Absolutely! A negative common difference means the sequence is decreasing. For example: 10, 7, 4, 1, -2, -5,... has common difference $d = -3$. The nth term formula would be $a_n = 10 + (n - 1) \times (-3) = 13 - 3n$. Many real-world situations involve decreasing linear sequences, like a car slowing down at a constant rate or a battery draining at a steady pace.
Q: What's the difference between a linear sequence and a linear function?
They are very closely related! A linear sequence is a special type of linear function where the input (x-values) are positive integers (1, 2, 3, 4,...). A linear function like $f(x) = 3x - 1$ can accept any real number as input, but when we only use positive integers, we get the linear sequence 2, 5, 8, 11, 14,.... So every linear sequence corresponds to a linear function, but not every linear function produces a sequence unless we restrict the domain to positive integers.
Linear sequences are fundamental mathematical patterns that appear throughout mathematics and everyday life. Their defining characteristic - a constant difference between consecutive terms - makes them predictable and easy to work with. By mastering the nth term formula $a_n = a_1 + (n - 1)d$, you can quickly find any term in a linear sequence without listing all previous terms. Recognizing these patterns in real-world situations like saving money, plant growth, or temperature changes helps us make predictions and solve practical problems. Whether you're dealing with increasing, decreasing, or constant sequences, the principles remain the same: identify the common difference, and you've unlocked the pattern!
Footnote
[1] Arithmetic Progression (AP): The formal mathematical term for a linear sequence. It is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference. The study of arithmetic progressions includes finding individual terms, sums of terms, and various properties of these sequences.