Arithmetic Sequence Calculator Formula
Understand the math behind the arithmetic sequence calculator. Each variable explained with a worked example.
Formulas Used
Nth Term
nth_term = a1 + (n - 1) * dSum
sum = n / 2 * (2 * a1 + (n - 1) * d)Average
average = (a1 + a1 + (n - 1) * d) / 2Variables
| Variable | Description | Default |
|---|---|---|
a1 | First Term (a₁) | 3 |
d | Common Difference (d) | 5 |
n | Number of Terms (n) | 10 |
How It Works
Arithmetic Sequence
nth Term
aₙ = a₁ + (n - 1) × d
Sum of n Terms
Sₙ = n/2 × (2a₁ + (n-1)d)
or equivalently: Sₙ = n/2 × (a₁ + aₙ)
An arithmetic sequence has a constant difference between consecutive terms.
Worked Example
Arithmetic sequence: first term = 3, common difference = 5, find the 10th term and sum.
- 01a₁₀ = 3 + (10-1) × 5 = 3 + 45 = 48
- 02Sum = 10/2 × (3 + 48) = 5 × 51 = 255
Frequently Asked Questions
What is an arithmetic sequence?
An arithmetic sequence has a constant difference between consecutive terms. Examples: 2,5,8,11,14... (d=3) or 100,90,80,70... (d=-10).
How do I find the common difference?
Subtract any term from the next term: d = a₂ - a₁. If this value is constant throughout, it is an arithmetic sequence.
What is the sum formula?
Sₙ = n/2 × (first + last). This is the number of terms times the average of the first and last terms.
Ready to run the numbers?
Open Arithmetic Sequence Calculator