3.12Arithmetic Sequences as Linear Functions
Every arithmetic sequence is a linear function in disguise. The common difference is the slope, the term number is the input, and the explicit rule is the equation — with a discrete domain of whole numbers.
Why This Matters
Arithmetic sequences appear on the SAT and ACT as pattern problems, function problems, and real-world modeling questions. Understanding how they connect to linear functions means you can solve sequence problems using algebra — and algebra problems using sequence reasoning.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
How Is an Arithmetic Sequence Related to a Linear Function?
An arithmetic sequence has a constant difference between consecutive terms. That constant difference acts like the slope of a linear function. Because sequence inputs are term numbers, its graph consists of separate points rather than a continuous line.
Lesson Overview
An arithmetic sequence is a list of numbers where each term is found by adding the same value to the previous term. That constant value is called the common difference. Because the common difference is constant, arithmetic sequences behave exactly like linear functions — with one key difference: the domain is discrete (only whole-number term positions), not continuous.
Identifying an Arithmetic Sequence
To check if a sequence is arithmetic, subtract consecutive terms. If the difference is the same every time, the sequence is arithmetic. If the differences vary, it is not arithmetic.
✓ Arithmetic
4, 7, 10, 13, 16, ...
7−4 = 3, 10−7 = 3, 13−10 = 3, 16−13 = 3
Constant difference → arithmetic. d = 3.
✗ Not Arithmetic
2, 4, 8, 16, 32, ...
4−2 = 2, 8−4 = 4, 16−8 = 8, 32−16 = 16
Differences are not equal → not arithmetic (geometric).
The Explicit Rule: aₙ = a₁ + (n−1)d
The explicit rule lets you find any term directly without listing all previous terms.
Formula Anatomy
a₁
First term of the sequence
n
Term number (position)
d
Common difference (slope)
This connects directly to f(x) = mx + b: the common difference d plays the role of slope, and a₁ − d plays the role of the y-intercept (the value at n = 0, which is not a real term in the sequence).
The Recursive Rule
A recursive rule defines each term using the previous term. It is useful for generating terms one at a time, but requires knowing the term before it.
You must always state the starting value a₁ — without it, the recursive rule is incomplete.
Explicit vs. Recursive Comparison
| Explicit Rule | Recursive Rule | |
|---|---|---|
| Formula | aₙ = a₁ + (n−1)d | a₁ = first term; aₙ = aₙ₋₁ + d |
| Use when | Finding any term directly | Generating terms in order |
| Example | Find a₁₀ without listing all terms | List the first 5 terms |
| Requires | a₁ and d only | Previous term aₙ₋₁ |
Arithmetic Sequences as Discrete Linear Functions
The explicit rule aₙ = a₁ + (n−1)d can be rewritten as aₙ = d·n + (a₁ − d), which has the form y = mx + b with m = d. This means every arithmetic sequence corresponds to a linear function — but with a key restriction: the domain is discrete (only positive integers), so the graph shows isolated dots, not a continuous line.
| n (term #) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| aₙ (value) | 3 | 7 | 11 | 15 | 19 |
Sequence: 3, 7, 11, 15, 19 (a₁=3, d=4). Explicit rule: aₙ = 4n − 1.
Sequence Table and Discrete Graph
Sequence 3, 7, 11, 15, 19 — Discrete Graph
Key observations:
- Isolated dots — no connecting line because n must be a whole number.
- The dashed line shows the linear trend (aₙ = 4n − 1) but is not part of the sequence graph.
- Each dot is exactly 4 units higher than the previous — the common difference d = 4.
- The domain is {1, 2, 3, 4, 5, ...} — discrete, not continuous.
Key Vocabulary
Arithmetic Sequence
A sequence where each term is found by adding a constant value (the common difference) to the previous term.
Example: 3, 7, 11, 15, 19, ... (common difference = 4)
Common Difference (d)
The constant value added to each term to get the next term. d = aₙ − aₙ₋₁.
Example: In 3, 7, 11, 15: d = 7−3 = 4.
Explicit Rule
A formula that gives the nth term directly: aₙ = a₁ + (n−1)d.
Example: For 3, 7, 11, 15: aₙ = 3 + (n−1)·4 = 4n − 1.
Recursive Rule
A formula that defines each term using the previous term: aₙ = aₙ₋₁ + d, with a₁ given.
Example: a₁ = 3, aₙ = aₙ₋₁ + 4.
Discrete Domain
A domain consisting of separate, distinct values — not a continuous range. For sequences, the domain is {'{1, 2, 3, 4, ...}'}.
Example: Term numbers 1, 2, 3, 4 are discrete — you cannot have term 2.5.
Math Tips
Use the explicit rule aₙ = a₁ + (n−1)d when you need to find a specific term without listing all previous terms.
The common difference d is the slope of the corresponding linear function. A positive d means increasing; negative d means decreasing.
Always check that the sequence is arithmetic before applying the explicit rule — verify that ALL consecutive differences are equal.
Worked Examples
Identify the common difference: 5, 9, 13, 17, 21, ...
Subtract consecutive terms: 9−5=4, 13−9=4, 17−13=4, 21−17=4.
The common difference is constant: d = 4.
This is an arithmetic sequence.
Find the 12th term of the sequence 2, 5, 8, 11, ...
Identify: a₁ = 2, d = 3.
Apply explicit rule: aₙ = a₁ + (n−1)d.
a₁₂ = 2 + (12−1)·3 = 2 + 33 = 35.
Write an explicit rule for the sequence 7, 3, −1, −5, ... (Negative common difference)
Identify: a₁ = 7, d = 3−7 = −4.
Apply: aₙ = 7 + (n−1)(−4).
Simplify: aₙ = 7 − 4n + 4 = −4n + 11.
Check: a₁ = −4(1)+11 = 7 ✓; a₂ = −4(2)+11 = 3 ✓
Write a recursive rule for the sequence 4, 4.5, 5, 5.5, ... (Fractional common difference)
Identify: a₁ = 4, d = 4.5−4 = 0.5.
Recursive rule: a₁ = 4; aₙ = aₙ₋₁ + 0.5 for n ≥ 2.
Check: a₂ = 4 + 0.5 = 4.5 ✓; a₃ = 4.5 + 0.5 = 5 ✓
A theater has 20 seats in the first row, 23 in the second, 26 in the third. Write an explicit rule for the number of seats in row n. How many seats are in row 15?
Identify: a₁ = 20, d = 3.
Explicit rule: aₙ = 20 + (n−1)·3 = 3n + 17.
Row 15: a₁₅ = 3(15)+17 = 45+17 = 62 seats.
Compare: Arithmetic sequence 3, 7, 11, 15, ... vs. linear function f(x) = 4x − 1. How are they related?
Sequence: a₁=3, d=4. Explicit rule: aₙ = 3+(n−1)·4 = 4n−1.
Linear function: f(x) = 4x−1.
They have the same formula! But the sequence is only defined for n=1,2,3,... (discrete), while f(x) is defined for all real x (continuous).
The graph of the sequence is isolated dots; the graph of f(x) is a continuous line.
Common Mistakes
Using aₙ = a₁ + n·d instead of aₙ = a₁ + (n−1)d.
The formula uses (n−1) because the first term already exists — you add d only (n−1) more times.
Confusing the term value with the term number.
n is the position (1st, 2nd, 3rd, ...); aₙ is the value at that position.
Drawing a continuous line for a sequence graph.
Sequence graphs show isolated dots because the domain is discrete — only whole-number term numbers.
Using a changing difference as if the sequence were arithmetic.
If the differences between consecutive terms are not all equal, the sequence is not arithmetic.
Forgetting to state a₁ when writing a recursive rule.
A recursive rule requires both the starting value (a₁) and the rule (aₙ = aₙ₋₁ + d).
Confusing common difference (arithmetic) with common ratio (geometric).
Arithmetic sequences ADD a constant; geometric sequences MULTIPLY by a constant.
Guided Practice
Guided Practice Video: Arithmetic Sequences as Linear Functions
Watch the guided practice walkthrough for arithmetic sequences as linear functions, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Is 2, 6, 18, 54, ... arithmetic? If so, find the common difference.
Hint: Subtract consecutive terms. Are all the differences equal?
Write an explicit rule for the sequence 10, 7, 4, 1, −2, ...
Hint: Find a₁ and d. Apply aₙ = a₁ + (n−1)d. Simplify.
A sequence has a₁ = 5 and d = 6. Find a₈ using the explicit rule.
Hint: Substitute a₁=5, d=6, n=8 into aₙ = a₁ + (n−1)d.
Interactive Practice — 3 Questions
What is the common difference of 5, 8, 11, 14, ...?
What is the 15th term of 2, 5, 8, 11, ...?
Which is the correct explicit rule for a₁=4, d=7?