Sequence Calculator
Enter a sequence's first term and common difference (arithmetic) or common ratio (geometric), along with which term you want, to find that term's value and the sum of the sequence up to that point.
How the Sequence formula works
Arithmetic and geometric sequence formulas:
Arithmetic nth term = a₁ + (n−1)d Arithmetic sum = n/2 × (2a₁ + (n−1)d) Geometric nth term = a₁ × r^(n−1) Geometric sum = a₁ × (rⁿ − 1) / (r − 1), for r ≠ 1
Step-by-step calculation
- For an arithmetic sequence, multiply the common difference by (n−1) and add the first term.
- For a geometric sequence, raise the common ratio to the power of (n−1) and multiply by the first term.
- Apply the corresponding sum formula to find the total of all terms up through the nth.
Worked example
Arithmetic sequence starting at 3 with common difference 4, finding the 10th term: 3 + (10−1)×4 = 3 + 36 = 39. Sum of first 10 terms = 10/2 × (2×3 + 9×4) = 5 × 42 = 210.
Frequently asked questions
What's the difference between arithmetic and geometric sequences?
An arithmetic sequence adds a constant amount (the common difference) between terms, while a geometric sequence multiplies by a constant ratio between terms — arithmetic grows linearly, geometric grows exponentially.
What happens if the geometric ratio is 1?
The sum formula for geometric sequences is undefined at r=1 (division by zero), since every term would simply equal the first term — in that case, the sum is just n × a₁.