Sequence Calculator

A mathematical sequence is an ordered list whose terms follow a rule. The position of a term matters: the first value is term one, the next value is term two, and so on. This sequence calculator generates the most common patterns used in school mathematics, finance, science, and computer programming. It also totals the displayed terms and evaluates a requested nth term, so you can check a pattern without writing every earlier value. An arithmetic sequence changes by a constant difference. Starting with 2 and adding 3 repeatedly gives 2, 5, 8, 11, and 14. Its nth term is found with a(n) = a(1) + (n - 1)d, where d is the common difference. Arithmetic patterns describe steady changes such as saving the same extra amount each month, regular salary increments, rows of seats, or distances measured at equal intervals. A negative difference produces a decreasing sequence just as easily. A geometric sequence changes by a constant ratio rather than a constant difference. Starting with 3 and multiplying by 2 gives 3, 6, 12, 24, and 48. The nth-term rule is a(n) = a(1) times r raised to the power n - 1. Geometric patterns model compound growth, depreciation, repeated scaling, radioactive decay, and many population models. Ratios between zero and one create decay, while negative ratios make signs alternate. The Fibonacci mode begins with zero and one, then creates each later term by adding the previous two. This famous recursive sequence appears in algorithms, counting problems, plant growth illustrations, and mathematical explorations. The custom mode accepts a list you already know. It is useful for totaling observations or locating a term without claiming that a unique formula has been inferred from limited data. Use the number-of-terms field to control the generated list. The calculator limits generated output to one hundred terms so results remain readable. The optional nth-term field can evaluate a position beyond the displayed list for arithmetic, geometric, and Fibonacci modes. Very large geometric or Fibonacci positions may exceed ordinary floating-point limits, so interpret unusually large scientific results carefully. For everyday exercises, the calculator provides a quick, transparent way to compare patterns, verify homework, and explore how changing a starting term, difference, or ratio changes an entire sequence.