Number Sequence Calculator

A number sequence is an ordered list of numbers that follow a fixed rule, where each value is called a term. This calculator generates the full list of terms, finds any specific term without listing the rest, and computes the running total (sum) for the two most important families of sequences: arithmetic and geometric. It is built for school and college students studying progressions, for teachers preparing examples, and for anyone who needs a quick, accurate answer to a sequence problem — from competitive-exam preparation to everyday financial reasoning.

Sequences appear far beyond the classroom. Arithmetic progressions model anything that grows by a fixed step — a recurring deposit of the same amount each month, a salary that rises by a flat increment, or seats arranged row by row in an auditorium. Geometric progressions model anything that grows or shrinks by a fixed percentage — compound interest, population growth, and depreciation of a vehicle. Recognising which type you are dealing with is the first and most important step.

Arithmetic Sequences

In an arithmetic sequence each term is obtained by adding a constant called the common difference (d) to the previous term. The difference between any two consecutive terms is always the same. The nth term is found with aₙ = a₁ + (n−1)×d, where a₁ is the first term. The sum of the first n terms is Sₙ = n/2 × (2a₁ + (n−1)d), which is equivalent to Sₙ = n/2 × (a₁ + aₙ) — half the count multiplied by the sum of the first and last terms. Example: 2, 5, 8, 11, 14… has a₁ = 2 and d = 3.

Geometric Sequences

In a geometric sequence each term is obtained by multiplying the previous term by a constant called the common ratio (r). The nth term is aₙ = a₁ × r^(n−1), and the sum of the first n terms is Sₙ = a₁(1−rⁿ)/(1−r) for any r ≠ 1 (when r = 1 every term is equal, so Sₙ = a₁ × n). Example: 3, 6, 12, 24, 48… has a₁ = 3 and r = 2. Because each step multiplies, geometric sequences grow (or decay) far faster than arithmetic ones — this is exactly why compound interest outpaces simple interest over time.

Worked example

Suppose an employee starts on a monthly salary of ₹30,000 and receives a flat raise of ₹2,500 every month (an arithmetic sequence with a₁ = 30,000 and d = 2,500). The salary in the 12th month is a₁₂ = 30,000 + (12−1)×2,500 = 30,000 + 27,500 = ₹57,500. The total earned over the 12 months is S₁₂ = 12/2 × (30,000 + 57,500) = 6 × 87,500 = ₹5,25,000. With this calculator you simply enter the first term, the common difference, and 12 terms to read off both results instantly, along with every intermediate month.

Tips for using this calculator

Switch between the Arithmetic and Geometric tabs to match your problem. Enter the first term exactly as given — if a question numbers terms from zero, shift it so your first listed value is term 1. For a decreasing arithmetic sequence, use a negative common difference. For a shrinking geometric sequence (such as depreciation), use a ratio between 0 and 1, like 0.9 for a 10% yearly fall. The tool caps output at 50 terms, which keeps the page fast and covers almost every textbook problem.

Common mistakes to avoid

The most frequent error is using n instead of n−1 in the nth-term formula — remember the first term already "uses up" one step, so the 10th term involves only 9 differences. A second mistake is confusing the two families: if you find a constant difference the sequence is arithmetic, but if you find a constant ratio it is geometric. Finally, the geometric sum formula divides by (1−r), so it is undefined at r = 1; in that special case simply multiply the first term by the number of terms.