Enter a sequence's first term and common difference or ratio to find its nth term and the sum of its first n terms.
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In an arithmetic sequence, each term adds a fixed common difference to the previous term: nth term = first term + (n − 1) × common difference. In a geometric sequence, each term multiplies the previous term by a fixed common ratio: nth term = first term × ratio^(n − 1). The sum of terms uses the standard arithmetic or geometric series sum formula.
Arithmetic: first term 2, common difference 3, 10 terms: the sequence runs 2, 5, 8, 11, 14, 17, 20, 23, 26, 29. The 10th term = 2 + (10−1)×3 = 2 + 27 = 29, and the sum of all 10 terms = (10÷2)×(2+29) = 5×31 = 155.
Geometric: first term 2, common ratio 3, 6 terms: the sequence runs 2, 6, 18, 54, 162, 486. The 6th term = 2 × 3⁵ = 2 × 243 = 486, and the sum of all 6 terms = 2×(3⁶−1)÷(3−1) = 2×728÷2 = 728.
Using the geometric formula's ratio exponent incorrectly, the nth term uses ratio^(n−1), not ratio^n, since the first term itself counts as the 0th multiplication by the ratio.
Picking the wrong sequence type for the pattern, if consecutive terms share a constant difference, it's arithmetic, if they share a constant ratio (multiplier), it's geometric, mixing these up gives a completely different nth term and sum.
Entering a common ratio of exactly 1 for a geometric sequence and expecting the standard sum formula to work, that formula divides by (ratio − 1), which is undefined at ratio 1, this calculator handles that case separately since every term is simply equal to the first term.
An arithmetic sequence adds the same amount each step (like 2, 5, 8, 11...). A geometric sequence multiplies by the same factor each step (like 2, 6, 18, 54...).
Yes, a negative common difference produces a decreasing arithmetic sequence, and a negative common ratio produces a geometric sequence that alternates sign.
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