Sequence Calculator
Calculate arithmetic, geometric, Fibonacci, and other mathematical sequences
Calculate Sequence Terms
Quick Examples
Sequence Terms
Formula:
aₙ = a₀ + n × d = 1 + n × 1
Step-by-step Calculation:
Given: First term a₀ = 1, Common difference d = 1
Formula: aₙ = a₀ + n × d
For position 0: a0 = 1 + 0 × 1
a0 = 1 + 0 = 1
Pattern Analysis
Example: Arithmetic Sequence
Find the 10th term of: 3, 7, 11, 15...
Step 1: Identify the pattern - Common difference d = 7 - 3 = 4
Step 2: First term a₀ = 3
Step 3: Use formula aₙ = a₀ + n × d
Step 4: a₁₀ = 3 + 10 × 4 = 3 + 40 = 43
Sequence Types
Arithmetic
Constant difference between terms
Example: 2, 5, 8, 11...
Geometric
Constant ratio between terms
Example: 3, 6, 12, 24...
Fibonacci
Each term = sum of previous two
Example: 0, 1, 1, 2, 3, 5...
Square Numbers
Perfect squares: n²
Example: 0, 1, 4, 9, 16...
Quick Facts
Prime numbers have only two divisors: 1 and themselves
Triangular numbers can form triangular patterns
Fibonacci sequence appears frequently in nature
Powers of 2 are fundamental in computer science
Understanding Mathematical Sequences
What is a Sequence?
A mathematical sequence is an ordered list of numbers where each number has a specific position. The position determines the value according to a specific rule or pattern.
Common Sequence Types
- •Arithmetic: Add same number each time
- •Geometric: Multiply by same number each time
- •Fibonacci: Sum of previous two terms
- •Figurate: Numbers that form geometric shapes
Key Formulas
Arithmetic Sequence
aₙ = a₀ + n × d
Geometric Sequence
aₙ = a₀ × rⁿ
Triangular Numbers
Tₙ = n(n+1)/2
Star Numbers
sₙ = 6n(n-1) + 1