Sequence Solver Calculator
An arithmetic sequence adds a constant common difference d to get each next term: aₙ = a₁ + (n−1)d. A geometric sequence multiplies by a constant common ratio r instead: aₙ = a₁·r^(n−1). Choose the sequence type, enter the first term, the common difference or ratio, and which term number n you want, to get both the nth term and the sum of the first n terms.
Arithmetic Sequences: Constant Difference
Each term adds the same common difference d to the one before it: aₙ = a₁ + (n−1)d. Take a₁=3, d=5: the 10th term is 3 + 9×5 = 48, and the sum of the first 10 terms is S₁₀ = 10/2×(6+45) = 255, ten terms averaging 25.5 each.
Geometric Sequences: Constant Ratio
Each term multiplies the one before it by a common ratio r: aₙ = a₁·r^(n−1). Take a₁=2, r=3: the 5th term is 2×3⁴=162, and the sum of the first 5 terms is S₅ = 2×(1−3⁵)/(1−3) = 242. When r=1 every term is identical, so the usual sum formula divides by zero and the sum simply becomes n×a₁ instead.
Recognizing Which Type You Have
Subtract consecutive terms: if the difference is constant, it's arithmetic. Divide consecutive terms instead: if the ratio is constant, it's geometric. A sequence can't be both unless it's constant (d=0 and r=1 simultaneously), most real sequences are clearly one or the other once you check.
Frequently Asked Questions
What's the difference between an arithmetic and a geometric sequence?
An arithmetic sequence has a constant difference between consecutive terms, add the same number each time, while a geometric sequence has a constant ratio, multiply by the same number each time. 2, 5, 8, 11 is arithmetic (d=3); 2, 6, 18, 54 is geometric (r=3).
What's the formula for the sum of an arithmetic sequence?
Sₙ = n/2 × (2a₁ + (n−1)d), equivalent to n times the average of the first and last term. For 3, 8, 13, 18, 23 (a₁=3, d=5, n=5): S₅ = 5/2×(6+20) = 65.
What's the formula for the sum of a geometric sequence?
Sₙ = a₁(1 − rⁿ)/(1 − r) when r ≠ 1, or simply Sₙ = n·a₁ when r = 1, since every term is identical. For 2, 6, 18, 54 (a₁=2, r=3, n=4): S₄ = 2×(1−81)/(1−3) = 80.
Can the common ratio be negative or a fraction?
Yes. A negative ratio alternates signs each term, like 3, −6, 12, −24 (r=−2). A ratio between −1 and 1, excluding 0, shrinks the sequence toward zero, like 100, 50, 25, 12.5 (r=0.5).
What happens if n is 1?
The nth term equals the first term for any sequence type when n=1, since there have been zero steps of adding d or multiplying by r yet. The sum of the first 1 term is also just a₁.