Mathematical Progression

All math tools

Analyze Arithmetic and Geometric progressions, find n-th terms, and calculate sums.

Input Parameters

Progression Type

Initial Term (a₁)

Common Difference (d)

Target Term (n)

Sequence Analysis

19.00

n-th Term (aₙ)

Calculated progression values

Sum of n Terms (Sₙ)100.00

Term Number10

About this calculator

Quick Reference

Arithmetic Sumn/2(a₁+aₙ)

Geometric Suma(1-rⁿ)/(1-r)

Infinite GP Suma/(1-r) if |r|<1

What is a Progression?

A mathematical progression is a sequence of numbers where each term is related to the previous one by a fixed rule. In an Arithmetic Progression (AP), you add a constant 'd'. In a Geometric Progression (GP), you multiply by a constant 'r'. These patterns are the basis for interest calculations, wave behavior, and natural growth patterns.

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Pro Tips

Common Difference (d): In AP, this is the amount you add. It can be negative.
Common Ratio (r): In GP, this is the multiplier. If |r| < 1, the infinite sum converges.
N-th Term: Always use (n-1) because the first term doesn't receive the first addition/multiplication.

Fun Facts

"The legend of the Wheat and Chessboard is a famous example of the explosive growth of a geometric progression."
"Arithmetic progressions are used in the design of everything from engine pistons to the structure of musical scales."

Formula: AP: aₙ = a₁ + (n-1)d | GP: aₙ = a₁·rⁿ⁻¹