Understanding Number Sequences

A number sequence is an ordered list of numbers generated by following a specific mathematical rule. The individual numbers within a sequence are referred to as "terms" (usually denoted as a₁, a₂, a₃, etc., where the subscript represents the position in the sequence). Understanding sequences is a fundamental part of algebra, calculus, and computer science.

Arithmetic Sequences

An arithmetic sequence is one of the simplest forms of mathematical sequences. It is created by adding (or subtracting) a constant value to the previous term. This constant value is known as the "common difference" (d).

• Example: 5, 10, 15, 20...
• First Term (a₁): 5
• Common Difference (d): +5
• Formula: aₙ = a₁ + (n - 1)d

Arithmetic sequences always graph as a perfectly straight line, representing linear growth or decay.

Geometric Sequences

A geometric sequence is created by multiplying (or dividing) the previous term by a constant value. This constant multiplier is known as the "common ratio" (r).

• Example: 2, 6, 18, 54...
• First Term (a₁): 2
• Common Ratio (r): 3 (because 2 × 3 = 6, 6 × 3 = 18)
• Formula: aₙ = a₁ × r^(n - 1)

Geometric sequences graph as a curve, representing exponential growth or decay. Compound interest in finance is a real-world example of a geometric sequence.

The Fibonacci Sequence

The Fibonacci sequence is a unique recursive sequence where each term is the sum of the two preceding terms. It generally begins with 0 and 1.

• Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...
• Formula: Fₙ = Fₙ₋₁ + Fₙ₋₂

This sequence appears frequently in nature, from the arrangement of leaves on a stem to the spiral of a nautilus shell, and is closely tied to the "Golden Ratio."

Series vs. Sequence

While a sequence is a list of numbers, a series is the sum of that list. Our calculator automatically generates the sequence based on your parameters and then calculates the finite series (the sum of all generated terms) to save you manual addition time.