In more detail
An arithmetic sequence adds a common difference d: the nth term is a₁ + (n − 1) × d and the sum of the first n terms is n × (first + last) / 2. A geometric sequence multiplies by a common ratio r: the nth term is a₁ × rⁿ⁻¹ and the sum of n terms is a₁ × (rⁿ − 1) / (r − 1). Other rules build each term from earlier ones, as in the Fibonacci sequence 1, 1, 2, 3, 5, 8…, where the ratio of neighbouring terms approaches 1.618.
Why it matters
Arithmetic sequences describe steady changes such as fixed monthly savings or seat rows; geometric ones describe compound growth, depreciation and halving. A few terms never prove the rule: 1, 2, 4 could continue with 8 (doubling) or 7 (adding 1, then 2, then 3), so "find the next term" puzzles need the rule stated.
Example
arithmetic 3, 7, 11, … d = 4
10th term = 3 + 9 × 4 = 39
sum of 10 = 10 × (3 + 39) / 2 = 210
geometric 2, 6, 18, … r = 3
6th term = 2 × 3⁵ = 486
sum of 6 = 2 × (3⁶ − 1) / (3 − 1) = 728
1 + 2 + … + 100 = 100 × 101 / 2 = 5,050