Arithmetic Sequence Calculator

An arithmetic sequence is an ordered list of numbers in which the difference between consecutive terms remains constant. That constant step is called the common difference. For example, 2, 5, 8, 11, 14 is arithmetic because each term is three more than the preceding term. A negative common difference creates a decreasing sequence, while a difference of zero creates a constant sequence. The nth term formula is aₙ = a₁ + (n - 1)d. The first term is a₁, d is the common difference, and n is the position to find. The subtraction of one matters because no difference has been added at the first position. At the fifth position of a sequence starting at 2 with difference 3, the formula gives 2 + 4 × 3 = 14. This direct formula avoids generating every earlier term. A finite arithmetic series is the sum of a specified number of sequence terms. Its sum can be found with Sₙ = n(a₁ + aₙ) / 2. This works by pairing the first and last terms, then the second and next-to-last terms; every pair has the same total. An equivalent formula is Sₙ = n(2a₁ + (n - 1)d) / 2. The calculator first determines the last term and then uses the first version to report the sum. Arithmetic patterns appear in savings plans with fixed deposits, seating rows that grow by a constant amount, evenly spaced measurements, linear depreciation, exercise schedules, and many algebra problems. They differ from geometric sequences, where terms are multiplied by a common ratio rather than increased by a common difference. Checking subtraction between adjacent terms is the fastest way to identify an arithmetic sequence. The first term and common difference may be positive, negative, zero, or decimal values. The number of terms must be a positive whole number because it counts positions. This tool accepts up to 1000 terms and displays the first ten as a preview, while still calculating the requested nth term and complete sum. A negative difference is useful for countdowns and steady decreases, such as 20, 16, 12, 8, 4, 0. To recover a common difference from two consecutive terms, subtract the earlier term from the later one. If nonconsecutive terms are known, subtract their values and divide by the difference in their positions. To solve for the number of terms, rearrange the nth-term equation, provided the difference is not zero. These transformations all come from the same linear relationship between term value and position. Decimal answers are rounded only for display. For money or measured data, apply the rounding convention appropriate to the task after calculating. The sequence model assumes an exactly constant difference; if observed data only approximately follows that pattern, an arithmetic sequence may describe a trend but will not capture random variation.