Common sequence types
An arithmetic sequence has a constant difference between consecutive terms: 2, 5, 8, 11 has a common difference of 3. A geometric sequence has a constant ratio: 3, 6, 12, 24 has a common ratio of 2. The Fibonacci sequence adds the two previous terms to generate the next: 1, 1, 2, 3, 5, 8, 13.
Polynomial sequences arise when the differences between terms are not constant but the second differences (or higher) are. The sequence of squares 1, 4, 9, 16, 25 has first differences 3, 5, 7, 9 and constant second differences of 2, confirming it follows a quadratic pattern.
Arithmetic: a_n = a_1 + (n - 1) x d
The n-th term of an arithmetic sequence with first term a_1 and common difference d.
Geometric: a_n = a_1 x r^(n - 1)
The n-th term of a geometric sequence with first term a_1 and common ratio r.