Number Sequence Calculator
Solve arithmetic, geometric, and Fibonacci progressions. Input terms to auto-detect sequence formulas and predict future intervals.
Term at Index a_n (n = 10)
19
Series Sum S_n
100
Progression Rate of Growth Curve
Rate curve plots indices up to first 15 values relative to peak heights. Negative values are color-coded red.
Step-by-step Mathematical Derivations
Arithmetic vs. Geometric Progressions
Progressions are ordered lists of numbers where terms follow a specific algebraic relationship:
- Arithmetic Progression (AP): Each term is obtained by adding a constant difference d to the preceding term. For example: 3, 7, 11, 15, ... (d = 4).
- Geometric Progression (GP): Each term is obtained by multiplying the preceding term by a constant ratio r. For example: 3, 6, 12, 24, ... (r = 2).
Finding Patterns with Auto-Detection
If you have an unknown list of numbers (such as 2, 6, 12, 20, 30), the auto-detect engine evaluates whether the terms increase linearly, geometrically, or follow a quadratic growth model (an = a n2 + b n + c). It computes differences between terms to determine the characteristic governing formula.
Frequently Asked Questions About Sequences
What is the difference between an arithmetic and a geometric progression?
An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant (e.g., 2, 4, 6, 8... where difference is 2). A geometric progression (GP) is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (e.g., 3, 9, 27, 81... where the ratio is 3).
When does an infinite geometric series converge?
An infinite geometric series converges if and only if the absolute value of the common ratio (r) is strictly less than 1 (|r| < 1). Under this condition, the infinite sum approaches a finite limit solved by the formula S = a1 / (1 - r).
How does the auto-detect feature recognize sequence patterns?
By evaluating the differences or ratios between consecutive terms. If differences are constant, it is arithmetic; if ratios are constant, it is geometric; if second-order differences are constant, it is quadratic; and if it matches F(n) = F(n-1) + F(n-2), it is Fibonacci-like.
Can this auto-detect sequence patterns?
Yes. Solve arithmetic, geometric, Fibonacci sequences and auto-detect patterns from numbers.