Generate and analyze arithmetic, geometric, Fibonacci, and other number sequences. Find nth terms, sums, and patterns.
Sequence Sum
100
Sum of 10 terms
Common difference (d) = 3 - 1 = 2
1, 3, 5, 7, 9, 11, 13, 15, 17, 19
First Term
1
Last Term (a_10)
19
Sum
100
Average
10.00
a_n = 1 + (n-1) × 2
a_n = a_1 + (n-1)d where a_1 = 1, d = 2
S_n = n/2 × (2a_1 + (n-1)d) = 10/2 × (2×1 + (10-1)×2)
| 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 |
2, 5, 8, 11, 14, ... (d=3)
a_n = a_1 + (n-1)d
3, 6, 12, 24, 48, ... (r=2)
a_n = a_1 × r^(n-1)
1, 1, 2, 3, 5, 8, 13, ...
F_n = F_(n-1) + F_(n-2)
1, 3, 6, 10, 15, 21, ...
T_n = n(n+1)/2
1, 4, 9, 16, 25, 36, ...
S_n = n²
2, 3, 5, 7, 11, 13, ...
No simple formula
Sequence Sum
100
Sum of 10 terms
Quick-start with common scenarios
Test your skills with practice problems
Practice with 4 problems to test your understanding.
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.
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.
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.
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 Sum
100
Sum of 10 terms