Arithmetic Sequences
Every term is the previous term plus a constant. That simple rule generates the nth term formula, the sum formula, and a deep connection to linear functions — all covered here.
Quick Reference — All Formulas
| Formula | Expression |
|---|---|
| nth term | aₙ = a₁ + (n − 1)d |
| nth term from middle | aₙ = aₖ + (n − k)d |
| Common difference | d = aₙ₊₁ − aₙ |
| d from first & nth | d = (aₙ − a₁) / (n − 1) |
| Sum (first + last) | Sₙ = n/2 · (a₁ + aₙ) |
| Sum (a₁ and d) | Sₙ = n/2 · (2a₁ + (n−1)d) |
Definition
An arithmetic sequence is a sequence where each term equals the previous term plus a fixed constant called the common difference d.
Increasing (d > 0)
3, 10, 17, 24, 31, ...
d = 7 (add 7 each time)
Decreasing (d < 0)
50, 45, 40, 35, 30, ...
d = −5 (subtract 5 each time)
The key test:
Subtract any term from the next: d = aₙ₊₁ − aₙ. If you get the same number every time, it is an arithmetic sequence.
The Formulas — Where They Come From
nth Term: aₙ = a₁ + (n−1)d
Start at a₁ and add d exactly (n−1) times (not n times — you do not add d on the first step).
nth Term from Any Known Term: aₙ = aₖ + (n−k)d
If you know the 5th term but not the 1st, you can still find any term. Just count the steps from position k to position n.
Example: a₅ = 23, d = 4 → a₁₂ = 23 + (12−5)(4) = 23 + 28 = 51
Sum Formula: Sₙ = n/2 · (a₁ + aₙ)
Gauss's insight: pair the first and last terms, the second and second-to-last, and so on. Every pair sums to the same value. There are n/2 such pairs.
When you do not know aₙ, substitute the nth term formula: Sₙ = n/2 · (2a₁ + (n−1)d)
Finding the Common Difference d
From consecutive terms
d = aₙ₊₁ − aₙ
Pick any two consecutive terms and subtract. Works when the sequence is listed explicitly.
From first and nth term
d = (aₙ − a₁) / (n − 1)
When only a₁ and aₙ are given. Rearranges the nth term formula.
4 Types of Problems
Given: a₁, d, and n
Plug into aₙ = a₁ + (n−1)d directly
Given: a₁, d, and aₙ
Solve aₙ = a₁ + (n−1)d for n algebraically
Given: a₁, d, and n
Use Sₙ = n/2 · (2a₁ + (n−1)d)
Given: aₙ, d, and n
Rearrange: a₁ = aₙ − (n−1)d
Arithmetic Means
Arithmetic means are the evenly spaced values inserted between two terms to create an arithmetic sequence. If you insert k arithmetic means between two values a and b, you create a sequence of (k + 2) terms total.
Method
- The total number of terms is k + 2 (k inserted means + the 2 endpoints).
- Find d using: d = (b − a) / (k + 1)
- Build the sequence by adding d repeatedly.
Example
Insert 3 arithmetic means between 4 and 24.
The three arithmetic means are 9, 14, and 19.
4 Worked Examples
Problem: a₁ = 3, d = 7. Find a₂₀.
Step 1: Identify a₁ = 3, d = 7, n = 20
Step 2: Apply the formula: aₙ = a₁ + (n−1)d
a₂₀ = 3 + (20 − 1)(7)
a₂₀ = 3 + 19 · 7
a₂₀ = 3 + 133 = 136
Problem: a₁ = 3, d = 5. Which term equals 103?
Step 1: Set aₙ = 103 and solve for n
103 = 3 + (n − 1)(5)
Step 2: Subtract 3 from both sides
100 = (n − 1)(5)
Step 3: Divide by 5
20 = n − 1
n = 21
Check: a₂₁ = 3 + 20(5) = 3 + 100 = 103 ✓
Problem: Find the sum of the first 50 terms of 2 + 5 + 8 + 11 + ...
Step 1: Identify a₁ = 2, d = 3, n = 50
Step 2: Find a₅₀ first
a₅₀ = 2 + (50 − 1)(3) = 2 + 147 = 149
Step 3: Apply the sum formula
S₅₀ = 50/2 · (a₁ + a₅₀)
S₅₀ = 25 · (2 + 149)
S₅₀ = 25 · 151 = 3,775
Alternate: S₅₀ = 50/2 · (2(2) + 49(3)) = 25 · (4 + 147) = 25 · 151 = 3,775 ✓
Problem: A person saves $50 in month 1, $75 in month 2, $100 in month 3, increasing by $25 each month. Find total savings after 12 months.
Step 1: Recognize the arithmetic sequence: a₁ = 50, d = 25, n = 12
Step 2: Find savings in month 12
a₁₂ = 50 + (12 − 1)(25) = 50 + 275 = 325
Step 3: Find total savings (sum of first 12 terms)
S₁₂ = 12/2 · (50 + 325)
S₁₂ = 6 · 375 = $2,250
The person saves $2,250 total over the 12 months.
Connection to Linear Functions
An arithmetic sequence is exactly a linear function restricted to positive integers. Compare the two formulas side by side:
Arithmetic Sequence
aₙ = a₁ + (n−1)d
= d·n + (a₁ − d)
Linear Function
f(x) = mx + b
The common difference d is the slope. Each step in n increases the value by d — exactly how slope measures rise per unit run.
Example: the sequence 3, 7, 11, 15, ... corresponds to the line f(x) = 4x − 1. Check: f(1) = 3, f(2) = 7, f(3) = 11. ✓ The slope is 4, the common difference is 4.
Frequently Asked Questions
What is an arithmetic sequence?
An arithmetic sequence is a list of numbers where each term is obtained by adding a fixed constant — called the common difference d — to the previous term. Example: 3, 7, 11, 15, 19, ... has d = 4. If d is positive the sequence increases; if d is negative it decreases. The key test: subtract any term from the next term and get the same number every time.
What is the formula for the nth term of an arithmetic sequence?
The nth term formula is aₙ = a₁ + (n−1)d, where a₁ is the first term, d is the common difference, and n is the term number. Example: for the sequence 3, 7, 11, ... with a₁ = 3 and d = 4, the 20th term is a₂₀ = 3 + (20−1)(4) = 3 + 76 = 79. You can also find a term from any known term using aₙ = aₖ + (n−k)d.
How do you find the sum of the first n terms of an arithmetic sequence?
The sum of the first n terms is Sₙ = n/2 · (a₁ + aₙ) when you know the first and last terms, or Sₙ = n/2 · (2a₁ + (n−1)d) when you know the first term and common difference. Example: sum of the first 50 terms of 2 + 5 + 8 + 11 + ... (a₁ = 2, d = 3): a₅₀ = 2 + 49(3) = 149, then S₅₀ = 50/2 · (2 + 149) = 25 · 151 = 3775.
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