Number Sequence Calculator

Arithmetic Sequences

An arithmetic sequence (or arithmetic progression) is a sequence where each term after the first is obtained by adding a constant value called the common difference d. The general term formula is a_n = a_1 + (n-1)d where a_1 is the first term and n is the term number. The sum of the first n terms is S_n = (n)/(2)(a_1 + a_n) = (n)/(2)(2a_1 + (n-1)d). Examples include: 2, 5, 8, 11, 14... (d=3) and 100, 95, 90, 85... (d=-5).

a_n = a_1 + (n-1)d \quad S_n = (n)/(2)(2a_1 + (n-1)d)

Geometric Sequences

A geometric sequence (or geometric progression) is a sequence where each term is obtained by multiplying the previous term by a constant value called the common ratio r. The general term is a_n = a_1 · r^n-1. The sum of the first n terms is S_n = a_1 · (1-r^n)/(1-r) for r ≠ 1. For infinite series where |r| < 1, the sum converges to S_∞ = (a_1)/(1-r). Examples: 3, 6, 12, 24... (r=2) and 64, 32, 16, 8... (r=0.5).

a_n = a_1 · r^n-1 \quad S_n = a_1 · (1-r^n)/(1-r) \text for r ≠ 1

Fibonacci Sequence

The Fibonacci sequence is one of the most famous sequences in mathematics, where each term is the sum of the two preceding terms: F_n = F_n-1 + F_n-2. Typically starting with F_1 = 1, F_2 = 1, the sequence proceeds: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... This sequence appears throughout nature (spiral patterns in shells, flower petals) and relates to the golden ratio \phi = (1+√(5))/(2) ≈ 1.618. The ratio of consecutive Fibonacci numbers \fracF_n+1F_n approaches \phi as n increases.

F_n = F_n-1 + F_n-2 \quad \lim_n \to ∞ \fracF_n+1F_n = \phi = (1+√(5))/(2)

Polygonal Number Sequences

Triangular numbers count objects arranged in equilateral triangles: T_n = (n(n+1))/(2), giving 1, 3, 6, 10, 15... Square numbers are perfect squares: S_n = n^2, giving 1, 4, 9, 16, 25... Pentagonal numbers follow P_n = (n(3n-1))/(2): 1, 5, 12, 22, 35... These figurate numbers have geometric interpretations and appear in number theory. Every positive integer is the sum of at most three triangular numbers (Gauss).

T_n = (n(n+1))/(2) \quad S_n = n^2 \quad P_n = (n(3n-1))/(2)

Power Sequences

Square sequences follow the pattern a_n = n^2: 1, 4, 9, 16, 25, 36... The differences between consecutive squares are the odd numbers: n^2 - (n-1)^2 = 2n-1. Cube sequences follow a_n = n^3: 1, 8, 27, 64, 125... and satisfy Σ_k=1^n k^3 = ((n(n+1))/(2))^2 = (T_n)^2. Higher powers like n^4, n^5 create rapidly growing sequences useful in polynomial analysis.

n^2 - (n-1)^2 = 2n-1 \quad Σ_k=1^n k^3 = (Σ_k=1^n k)^2

Recurrence Relations

Many sequences are defined recursively through recurrence relations, expressing each term as a function of previous terms. Linear recurrence relations have the form a_n = c_1 a_n-1 + c_2 a_n-2 + ·s + c_k a_n-k. The Fibonacci sequence is a second-order linear recurrence. Solving recurrence relations involves finding closed-form expressions. The characteristic equation method uses polynomial roots to find explicit formulas for sequences defined recursively.

a_n = c_1 a_n-1 + c_2 a_n-2 + ·s + c_k a_n-k

Worked Example

Inputs

• sequenceType: arithmetic
• firstTerm: 2
• commonDifference: 3
• terms: 5

Result: 2, 5, 8, 11, 14

Explanation

Arithmetic sequence starting with 2, common difference of 3

Second Scenario

Inputs

• sequenceType: arithmetic
• firstTerm: 2.4
• commonDifference: 3
• terms: 5

Result: 2, 5, 8, 11, 14

Explanation

This scenario uses different inputs (sequenceType = arithmetic, firstTerm = 2.4, commonDifference = 3, terms = 5) to show how changing one variable affects the number sequence result. Run the calculator above with these values to get the exact updated output with step-by-step work.