11.2Arithmetic Sequences
Master the explicit formula aₙ = a₁ + (n−1)d and recursive formula to find any term, insert arithmetic means, and solve real-world applications.
Arithmetic sequences model constant-rate change — salary increases, loan payments, and staircase designs all follow arithmetic patterns. Mastering them builds the foundation for linear functions and financial mathematics.
Essential Question
How can we use a constant difference to describe, extend, and apply arithmetic sequences in real-world situations?
Overview
An arithmetic sequence is a list of numbers where each term is obtained by adding a fixed value — the common difference d — to the previous term. Arithmetic sequences model constant-rate change: salary raises, staircase tile counts, and evenly spaced measurements.
- Common difference: d = aₙ − aₙ₋₁ (constant for all consecutive pairs)
- Explicit formula: aₙ = a₁ + (n − 1)d
- Recursive formula: aₙ = aₙ₋₁ + d, with a₁ given
- Finding d from two terms: d = (aₙ − a₁) / (n − 1)
- Arithmetic means: terms inserted evenly between two values
Worked Examples
Find the first 5 terms and the common difference for the sequence with a₁ = 7 and d = −3.
Start with a₁ = 7.
a₂ = 7 + (−3) = 4
a₃ = 4 + (−3) = 1
a₄ = 1 + (−3) = −2
a₅ = −2 + (−3) = −5
The common difference is d = −3 (each term decreases by 3).
Find a₁₀ for the arithmetic sequence with a₁ = 4 and d = 6 using the explicit formula.
Explicit formula: aₙ = a₁ + (n − 1)d
Substitute: a₁₀ = 4 + (10 − 1)(6)
a₁₀ = 4 + 9 · 6
a₁₀ = 4 + 54 = 58
Given a₃ = 11 and a₇ = 27, find d and write the explicit formula.
Use d = (aₙ − aₘ) / (n − m) with n=7, m=3.
d = (27 − 11) / (7 − 3) = 16 / 4 = 4
Find a₁: a₃ = a₁ + (3−1)(4) → 11 = a₁ + 8 → a₁ = 3
Explicit formula: aₙ = 3 + (n − 1) · 4
Simplify: aₙ = 4n − 1
Insert 3 arithmetic means between 5 and 25.
With 3 means inserted, there are 5 terms total: a₁ = 5, a₅ = 25.
Find d: d = (25 − 5) / (5 − 1) = 20 / 4 = 5
a₂ = 5 + 5 = 10
a₃ = 10 + 5 = 15
a₄ = 15 + 5 = 20
The 3 arithmetic means are 10, 15, 20.
A job starts at $42,000 with $2,500 annual raises. Find the salary in year 8.
This is arithmetic with a₁ = 42,000 and d = 2,500.
Use aₙ = a₁ + (n − 1)d with n = 8.
a₈ = 42,000 + (8 − 1)(2,500)
a₈ = 42,000 + 7 · 2,500
a₈ = 42,000 + 17,500 = 59,500
Guided Practice
Find the first 5 terms of the arithmetic sequence with a₁ = 2 and d = 5.
Hint: Add d = 5 repeatedly starting from a₁ = 2. Use the recursive formula: aₙ = aₙ₋₁ + 5.
Find a₁₂ for the arithmetic sequence with a₁ = −3 and d = 4.
Hint: Use the explicit formula aₙ = a₁ + (n − 1)d. Substitute n = 12, a₁ = −3, d = 4.
Find d and the explicit formula given a₂ = 9 and a₅ = 21.
Hint: Use d = (a₅ − a₂) / (5 − 2). Then find a₁ using a₂ = a₁ + d.
Insert 2 arithmetic means between 8 and 20.
Hint: With 2 means, there are 4 terms total: a₁ = 8, a₄ = 20. Find d = (20 − 8) / (4 − 1).
A theater has row 1 with 20 seats, and each row adds 3 seats. How many seats are in row 15?
Hint: This is arithmetic with a₁ = 20 and d = 3. Use aₙ = a₁ + (n − 1)d with n = 15.
Common Mistakes
Using n instead of (n − 1) in the explicit formula: aₙ = a₁ + n·d
The correct formula is aₙ = a₁ + (n − 1)d because the first term requires 0 additions of d.
Assuming a sequence is arithmetic after checking only one pair of consecutive terms.
Check that the difference is constant for ALL consecutive pairs before concluding it is arithmetic.
When inserting k arithmetic means, using k+1 as the number of intervals.
Inserting k means creates k+2 total terms and k+1 equal intervals, so d = (last − first)/(k+1).
Confusing the term number n with the term value aₙ.
n is the position (1, 2, 3, …); aₙ is the value at that position.
Math Tips
To verify a sequence is arithmetic, subtract consecutive terms — the differences must all be equal.
The explicit formula aₙ = a₁ + (n−1)d is equivalent to the slope-intercept form y = mx + b, where d is the slope.
A negative common difference means the sequence is decreasing.
When finding a₁ from a later term, work backwards: a₁ = aₙ − (n−1)d.
Arithmetic means divide an interval into equal parts — think of them like equally spaced points on a number line.
Arithmetic Sequence
A sequence where each term differs from the previous by a constant amount.
Common Difference (d)
The constant value added to each term to get the next: d = aₙ − aₙ₋₁.
Explicit Formula
A formula that gives the nth term directly: aₙ = a₁ + (n−1)d.
Recursive Formula
A formula that defines each term using the previous term: aₙ = aₙ₋₁ + d.
Arithmetic Means
Terms inserted between two values so that all terms form an arithmetic sequence.
Interactive Practice — 5 Questions
What is the common difference of the sequence 2, 9, 16, 23, 30?
Which formula correctly gives the nth term of an arithmetic sequence?
Find a₈ for the sequence with a₁ = 5 and d = −4.
How many arithmetic means are inserted if the total sequence has 6 terms?
Given a₂ = 7 and a₆ = 23, what is d?