Arithmetic Sequence Calculator

A sequence where each term increases by a fixed amount.

How the Math Works

An arithmetic sequence is a series of numbers where each term increases or decreases by a constant value called the common difference (d). The formula aₙ = a₁ + (n−1)d calculates any term's value by starting with the first term (a₁) and adding (n−1) multiples of the common difference. The sum formula Sₙ = n(a₁ + aₙ)/2 works by averaging the first and last term values and multiplying by the total number of terms, efficiently adding all terms without manual iteration.

Practical Applications

This calculator is essential in financial planning to determine cumulative savings with regular deposits or loan payments with fixed interest rates. Engineers use it to model uniform motion problems, like calculating total distance traveled at consistent speed intervals. In computer science, it helps analyze algorithms with linear time complexity by predicting operation counts across input sizes.

Day-to-Day Use

You can apply this when budgeting monthly expenses that increase by a fixed amount, like gym memberships or subscription fees. It's useful for tracking progress in goals such as reading a book page by page or saving for a vacation with incremental contributions. Even in sports training, it helps plan progressive workout intensities or competition schedules with evenly spaced intervals.

Worked example

2, 5, 8, … 10th term = 29, sum = 155.

FAQ