Find the nth term and the sum of the first n terms of an arithmetic or geometric sequence.
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The sequence and series calculator solves two basic types of mathematical sequences. An arithmetic sequence is one where each term increases over the previous one by a constant amount called the "common difference" (d), and its general term is calculated by the equation aₙ = a₁ + (n−1)d, while the sum of its first n terms is calculated by the equation Sₙ = (n/2)(2a₁ + (n−1)d). A geometric sequence is one where each term is multiplied by a constant ratio called the "common ratio" (r) to get the next term, and its general term is calculated by the equation aₙ = a₁ × rⁿ⁻¹, while the sum of its first n terms is calculated by the equation Sₙ = a₁(1−rⁿ)/(1−r). These sequences are used in calculating compound bank interest, population growth, physics, and many financial and scientific applications. Recognizing which type a given sequence belongs to is usually the first step: checking whether consecutive terms differ by a constant amount (arithmetic) or a constant ratio (geometric) immediately determines which pair of formulas actually applies.
Sequences and series appear early in most math curricula, but their real-world importance goes well beyond the classroom, since the two basic types covered here — arithmetic and geometric — describe two of the most fundamental patterns of change found throughout finance, biology, and physics.
An arithmetic sequence grows (or shrinks) by adding the exact same fixed amount at every single step — 2, 5, 8, 11, 14, each term exactly 3 more than the last. This pattern describes linear, steady change: a savings plan that deposits the same fixed amount every month, a staircase where every step rises by the same fixed height, or seating arranged in rows that each add the same fixed number of additional seats.
A geometric sequence, by contrast, grows (or shrinks) by multiplying by the exact same fixed ratio at every step — 3, 6, 12, 24, 48, each term exactly double the last. This pattern describes exponential change, and it grows dramatically faster than arithmetic growth over time, since each new term compounds on an already-larger previous term rather than simply adding a fixed amount to it.
Compound interest is the clearest financial example of a geometric sequence: a savings account earning a fixed interest rate each period grows geometrically, since the interest earned in one period is calculated on an already-larger balance than the period before, compounding the growth. This is precisely why compound interest eventually outpaces simple, arithmetic-style interest calculated only on the original principal, even though the two might look similar over a short time horizon.
Population growth is frequently modeled geometrically in its early, unconstrained stage (each generation multiplying by roughly the same growth factor), while many manufacturing and scheduling problems — evenly spaced production runs, evenly spaced maintenance intervals — follow arithmetic patterns instead. Physics uses arithmetic sequences to describe motion under constant acceleration at evenly spaced time intervals, and geometric sequences to describe radioactive decay and other exponential processes viewed at evenly spaced time steps.
The sum formulas for both sequence types matter just as much as the individual terms themselves — calculating the total interest earned across an entire loan term, or the total distance traveled across an entire accelerating motion, both require summing an entire sequence rather than evaluating any single term in isolation.
In an arithmetic sequence, consecutive terms differ by a fixed amount added each time, while in a geometric sequence, consecutive terms are multiplied by a fixed ratio.
It's the total you get by adding together every term in the sequence from the first one up to the nth term.
Yes, a negative ratio makes the sequence alternate in sign, and a fractional ratio (like 0.5) makes the terms shrink toward zero.