Sequences Calculator
Find the nth term, calculate the sum of a progression, and generate the next terms for arithmetic and geometric sequences instantly.
Calculate Your Sequence Properties
Enter the parameters of your arithmetic or geometric sequence to instantly find the nth term, the total sum, and the upcoming values.
Sequence Solver
Get step-by-step sequence calculations in seconds.
How it Works
Understanding sequence mathematics
Identify the Pattern
We determine if the sequence progresses by adding a constant (Arithmetic) or multiplying by a constant (Geometric).
Apply the Nth Term Formula
For arithmetic: aₙ = a + (n-1)d. For geometric: aₙ = a × r^(n-1). We plug in your values to find the specific term.
Calculate the Sum
We use the respective sum formulas: Sₙ = n/2[2a + (n-1)d] for arithmetic, or Sₙ = a(1 – r^n)/(1 – r) for geometric.
Generate Future Terms
Using the identified pattern, we project the next three consecutive terms in the sequence for your reference.
Essential Sequence Formulas
A quick reference guide for the most common arithmetic and geometric progression formulas used in GCSE and A-Level mathematics.
| Property | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Nth Term | aₙ = a + (n – 1)d | aₙ = a × r^(n – 1) |
| Sum of n Terms | Sₙ = n/2 × [2a + (n – 1)d] | Sₙ = a(1 – r^n) / (1 – r) [if r ≠ 1] |
| Sum to Infinity | Does not exist (diverges) | S_∞ = a / (1 – r) [only if |r| < 1] |
| Common Value | d = aₙ – aₙ₋₁ (Difference) | r = aₙ / aₙ₋₁ (Ratio) |
| Example | 3, 7, 11, 15… (a=3, d=4) | 2, 6, 18, 54… (a=2, r=3) |
Sequences & Series FAQ
Answers to the most frequently asked questions about arithmetic and geometric progressions in mathematics.
An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant is called the common difference (d). For example: 2, 5, 8, 11, where d = 3.
The nth term of a geometric sequence is found using the formula: aₙ = a × r^(n-1), where ‘a’ is the first term, ‘r’ is the common ratio, and ‘n’ is the position of the term in the sequence.
The sum of the first n terms of an arithmetic sequence is calculated using the formula: Sₙ = n/2 × [2a + (n – 1)d], where ‘a’ is the first term, ‘d’ is the common difference, and ‘n’ is the number of terms.
Yes, a common ratio can be negative. When the common ratio is negative, the terms of the geometric sequence will alternate between positive and negative values (e.g., 3, -6, 12, -24, where r = -2).