Introduction to Number Sequences and Progressions
A number sequence is an ordered list of numbers that follow a specific mathematical rule. The individual numbers in a sequence are called terms. Sequences are fundamental in algebra, calculus, and discrete mathematics, and they appear in natural patterns (like the Fibonacci sequence in plants) and financial models (like compound interest progressions). The sum of the terms of a sequence is called a series. Solving sequences involves finding the general formula for the $n$-th term, generating list values, and evaluating finite or infinite summations.
The two most common types of sequences are arithmetic sequences and geometric sequences. In an arithmetic sequence, the difference between consecutive terms is constant. In a geometric sequence, the ratio of consecutive terms is constant. Both progressions have unique formulas for finding terms and summing series, which are essential for engineering analysis, physics models, and financial planning.
This calculator generates sequences step-by-step. By entering the first term $a_1$, the common difference $d$ or ratio $r$, and the number of terms $n$, you can calculate any specific term, find the algebraic summation $S_n$, and examine whether infinite geometric series converge to a finite limit.