Sequences and Series Calculator
Choose a sequence type, enter the first term and common difference or ratio, and how many terms to find the nth term and the sum.
Last updated: September 5, 2026
Frequently asked questions
What's the formula for the nth term of an arithmetic sequence?
aₙ = a₁ + (n−1)d, where a₁ is the first term, d is the common difference, and n is the term number. For a₁=3, d=5, n=10: 3 + 9×5 = 48.
What's the formula for the sum of an arithmetic sequence?
Sum = n/2 × (2a₁ + (n−1)d), or equivalently n × (first term + last term) / 2. For the first 10 terms of 3, 8, 13...: 10/2 × (6 + 45) = 255.
What's the formula for the nth term of a geometric sequence?
aₙ = a₁ × r^(n−1), where r is the common ratio. For a₁=2, r=3, n=6: 2 × 3⁵ = 486.
What's the formula for the sum of a geometric sequence?
Sum = a₁(1 − rⁿ) / (1 − r) for r ≠ 1. For a₁=2, r=3, n=6: 2×(1−729)/(1−3) = 728. If r = 1, every term equals a₁, so the sum is simply a₁ × n.
What happens to a geometric series if the ratio r is between -1 and 1?
Each term shrinks toward zero, and while this calculator computes the sum of a finite number of terms n, an infinite geometric series with |r| < 1 would converge to a finite total of a1 / (1 − r).
Can the common difference or ratio be negative?
Yes - a negative common difference produces a decreasing arithmetic sequence (like 10, 7, 4, 1...), and a negative common ratio produces a geometric sequence that alternates sign each term (like 2, -6, 18, -54...).
Is a Fibonacci sequence the same as an arithmetic or geometric sequence?
No - a Fibonacci sequence adds the two previous terms to get the next one (1, 1, 2, 3, 5, 8...), which is neither a constant difference (arithmetic) nor a constant ratio (geometric), so this calculator doesn't cover it.