Number Sequence Calculator
Calculate the nth term and sum of arithmetic or geometric sequences. Enter the first term, the common difference (or ratio), and which term you want to find.
A number sequence is an ordered list of numbers that follows a specific pattern. The two most common types are arithmetic and geometric sequences.
Arithmetic Sequences
Each term differs from the previous by a constant amount called the common difference (d).
- nth term: a_n = a₁ + (n-1)d
- Sum of first n terms: S_n = n/2 x (a₁ + a_n) = n/2 x (2a₁ + (n-1)d)
Example: 3, 7, 11, 15, 19... has a₁ = 3, d = 4. The 10th term is 3 + 9(4) = 39. The sum of the first 10 terms is 10/2 x (3 + 39) = 210.
Geometric Sequences
Each term is multiplied by a constant ratio (r) to get the next term.
- nth term: a_n = a₁ x r^(n-1)
- Sum of first n terms: S_n = a₁(1 - r^n)/(1 - r), when r is not 1
Example: 2, 6, 18, 54... has a₁ = 2, r = 3. The 5th term is 2 x 3⁴ = 162. The sum of the first 5 terms is 2(1 - 243)/(1 - 3) = 242.
Identifying the Type
If the differences between consecutive terms are constant, it is arithmetic. If the ratios are constant, it is geometric. If neither, the sequence follows a different pattern.