The rule from one term to the next
A term-to-term rule says how to get the next term from the one before: 5, 9, 13, 17, ... is "add 4"; 3, 6, 12, 24, ... is "multiply by 2".
A rule can have two steps: start at 2, then "multiply by 3 and subtract 1" gives 2, 5, 14, 41, ...
Using the position: the nth term
The nth term works out any term from its position n. If the nth term is 3n + 4, the 1st term is 3 × 1 + 4 = 7, the 2nd is 10, and the 50th is 3 × 50 + 4 = 154: no need to list the first 49.
Matchstick squares in a row use 4, 7, 10, ... sticks: each new square adds 3, so pattern n has 3n + 1 sticks.