Number Sequence Calculator

An arithmetic sequence is a list of numbers where each term is obtained by adding a fixed value called the common difference (d) to the previous term. For example, 1, 3, 5, 7, 9 has a common difference of 2. The general formula is aₙ = a₁ + d × (n − 1).

A geometric sequence is one where each term is found by multiplying the previous term by a fixed number called the common ratio (r). For example, 2, 4, 8, 16 has a common ratio of 2. The formula is aₙ = a₁ × r^(n−1).

The Fibonacci sequence starts with 0 and 1, and each subsequent term is the sum of the two preceding terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... The rule is defined as aₙ = aₙ₋₁ + aₙ₋₂, with a₀ = 0 and a₁ = 1.

First, determine the type of sequence. For arithmetic, find the common difference and add it to the last term. For geometric, find the common ratio and multiply it by the last term. For Fibonacci, add the last two terms together. This calculator does this automatically once you enter the parameters.

The common difference (d) is the fixed amount added between consecutive terms in an arithmetic sequence. To find it, subtract any term from the one that follows it: d = aₙ − aₙ₋₁. For example, in 5, 10, 15, 20, the common difference is 5.

Yes. A negative common ratio produces an alternating sequence (e.g., 3, −6, 12, −24). A fractional ratio less than 1 produces a decreasing sequence (e.g., 16, 8, 4, 2, 1). This calculator supports any real-number ratio.

For arithmetic sequences, the sum of n terms is Sₙ = n/2 × (2a₁ + (n−1)d). For geometric sequences, Sₙ = a₁ × (1 − rⁿ) / (1 − r) when r ≠ 1. This calculator displays the cumulative sum for all shown terms in the table.

You can generate up to 50 terms at once using the 'Number of Terms to Show' field. The table displays all terms with their values and cumulative sums, paginated for easy reading.