Number Sequence Calculator
Generate and analyze arithmetic, geometric, Fibonacci, and other number sequences. Find nth terms, sums, and patterns.
Sequence Summary
Sequence Sum
100
Sum of 10 terms
Sequence Type
Parameters
Common difference (d) = 3 - 1 = 2
Generated Sequence
1, 3, 5, 7, 9, 11, 13, 15, 17, 19
First Term
1
Last Term (a_10)
19
Sum
100
Average
10.00
Sequence Visualization
Formulas
General Term (nth term)
a_n = 1 + (n-1) × 2
a_n = a_1 + (n-1)d where a_1 = 1, d = 2
Sum Formula
S_n = n/2 × (2a_1 + (n-1)d) = 10/2 × (2×1 + (10-1)×2)
Term Table
| n | Term (a_n) | Cumulative Sum | Difference |
|---|---|---|---|
| 1 | 1 | 1 | - |
| 2 | 3 | 4 | 2 |
| 3 | 5 | 9 | 2 |
| 4 | 7 | 16 | 2 |
| 5 | 9 | 25 | 2 |
| 6 | 11 | 36 | 2 |
| 7 | 13 | 49 | 2 |
| 8 | 15 | 64 | 2 |
| 9 | 17 | 81 | 2 |
| 10 | 19 | 100 | 2 |
Common Sequences Reference
Arithmetic
2, 5, 8, 11, 14, ... (d=3)
a_n = a_1 + (n-1)d
Geometric
3, 6, 12, 24, 48, ... (r=2)
a_n = a_1 × r^(n-1)
Fibonacci
1, 1, 2, 3, 5, 8, 13, ...
F_n = F_(n-1) + F_(n-2)
Triangular
1, 3, 6, 10, 15, 21, ...
T_n = n(n+1)/2
Square
1, 4, 9, 16, 25, 36, ...
S_n = n²
Prime
2, 3, 5, 7, 11, 13, ...
No simple formula
Sequence Summary
Sequence Sum
100
Sum of 10 terms
Sequence Examples
Quick-start with common scenarios
Practice Sequences
Test your skills with practice problems
Practice with 4 problems to test your understanding.
?How Do You Find Number Sequences?
Arithmetic sequence: each term differs by constant d. Formula: a_n = a_1 + (n-1)d. Sum: S_n = n(a_1 + a_n)/2. Geometric sequence: each term multiplied by constant r. Formula: a_n = a_1 * r^(n-1). Sum: S_n = a_1(1-r^n)/(1-r). Identify the pattern to determine sequence type.
What is a Number Sequence?
A number sequence is an ordered list of numbers following a specific pattern or rule. Arithmetic sequences have a constant difference between consecutive terms, while geometric sequences have a constant ratio. Understanding sequences is fundamental to algebra, calculus, finance, and many applications involving patterns and series.
Key Facts About Number Sequences
- Arithmetic: constant difference between terms (2, 5, 8, 11...)
- Geometric: constant ratio between terms (2, 6, 18, 54...)
- Arithmetic nth term: a_n = a_1 + (n-1)*d
- Geometric nth term: a_n = a_1 * r^(n-1)
- Arithmetic sum: S_n = n(a_1 + a_n)/2 = n(2a_1 + (n-1)d)/2
- Geometric sum: S_n = a_1(1-r^n)/(1-r) when r is not equal to 1
- Infinite geometric sum (|r|<1): S = a_1/(1-r)
- Common patterns: Fibonacci (add previous two), triangular numbers
Quick Answer
Arithmetic sequence: each term differs by constant d. Formula: a_n = a_1 + (n-1)d. Sum: S_n = n(a_1 + a_n)/2. Geometric sequence: each term multiplied by constant r. Formula: a_n = a_1 * r^(n-1). Sum: S_n = a_1(1-r^n)/(1-r). Identify the pattern to determine sequence type.
Frequently Asked Questions
An arithmetic sequence has a constant difference between consecutive terms. Formula: a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. Example: 2, 5, 8, 11, 14... has d = 3.
A geometric sequence has a constant ratio between consecutive terms. Formula: a_n = a_1 × r^(n-1), where a_1 is the first term and r is the common ratio. Example: 2, 6, 18, 54... has r = 3.
The Fibonacci sequence starts with 0, 1 (or 1, 1), and each subsequent term is the sum of the two preceding terms. F_n = F_(n-1) + F_(n-2). The sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21... appears throughout nature.
For arithmetic: a_n = a_1 + (n-1)d. For geometric: a_n = a_1 × r^(n-1). For Fibonacci: use recursion or Binet's formula. For triangular: n(n+1)/2. For squares: n². First identify the pattern, then apply the formula.
Arithmetic sum: S_n = n(a_1 + a_n)/2 or n(2a_1 + (n-1)d)/2. Geometric sum: S_n = a_1(1-r^n)/(1-r) for r≠1. Special sequences have their own formulas (e.g., squares: n(n+1)(2n+1)/6).
Last updated: 2025-01-15
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Sequence Summary
Sequence Sum
100
Sum of 10 terms