An arithmetic sequence grows by adding the same amount each step: 3, 7, 11, 15, … Savings with a fixed monthly deposit, seats in rows of a theatre and stacked pipes follow this pattern. The young Gauss famously added 1 to 100 in seconds by pairing terms: 50 pairs of 101 = 5,050.
How to use the Arithmetic Sequence Calculator
- What do you know?
- First term a₁
- Common difference d
- Position m
- Term aₘ
- Position k
- Term aₖ
- n (term to find / number of terms to add)
Results update as you type, and everything is calculated in your browser.
How it works
Choose what you know. From two terms aₘ and aₖ, the difference is d = (aₖ − aₘ)/(k − m) and the first term follows. The sum pairs the first and last terms.
Inputs explained
| Field | What to enter |
|---|---|
| What do you know? | Choose one: First term a₁ and difference d, Two terms of the sequence |
| First term a₁ | A number |
| Common difference d | A number |
| Position m | A number |
| Term aₘ | A number |
| Position k | A number |
| Term aₖ | A number |
| n (term to find / number of terms to add) | A number |
What each result means
- n-th term aₙ = [First term a₁] + ([n (term to find ÷ number of terms to add)] – 1)×[Common difference d]
- Sum of the first n terms Sₙ = [n (term to find ÷ number of terms to add)]×([First term a₁] + [n-th term aₙ])÷2
Example values
How the main result changes when First term a₁ changes and everything else stays at the example values above:
| First term a₁ | n-th term aₙ |
|---|---|
| 1.5 | 77.5 |
| 2.25 | 78.25 |
| 3 | 79 |
| 3.75 | 79.75 |
| 4.5 | 80.5 |
| 6 | 82 |
Formula
aₙ = a₁ + (n − 1)·d
Sₙ = n·(a₁ + aₙ)/2 = n·(2a₁ + (n − 1)d)/2
Worked example
Inputs: What do you know?: First term a₁ and difference d; First term a₁: 3; Common difference d: 4; n (term to find / number of terms to add): 20.
Results: Common difference d = 4; First term a₁ = 3; n-th term aₙ = 79; Sum of the first n terms Sₙ = 820.
Key terms
- Common difference
- The constant added to get the next term.
Frequently asked questions
How did Gauss add 1 to 100?
Pairing 1 + 100, 2 + 99, …: 50 pairs of 101 = 5,050.
Can d be negative?
Yes. The sequence then decreases.
How do I find d from two terms?
d = (aₖ − aₘ)/(k − m).
What is the difference from a geometric sequence?
Arithmetic adds a constant; geometric multiplies by one.