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
What is an arithmetic sequence?
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.
What is a geometric sequence?
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.
What is the Fibonacci sequence?
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.
How do I find the nth term?
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.
How do I find the sum of a sequence?
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: 2026-01-15
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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