Difference, then adjust
- 5, 8, 11, 14, ...
- the difference is 3, so start with 3n: 3, 6, 9, 12
- each term is 2 more than 3n
- nth term = 3n + 2
Check with n = 1: 3 + 2 = 5. A decreasing sequence has a negative difference: 20, 17, 14, ... is −3n + 23.
Is it in the sequence?
Set the nth term equal to the number and solve. Is 100 in 3n + 2? 3n + 2 = 100 gives n = 32.67, not a whole number, so no. Is 101? n = 33, so yes, it is the 33rd term.
Watch the sequence on a graph
Plot y = 3x + 2 and look at the points where x = 1, 2, 3, 4. The gradient is the common difference; the line's height at x = 0 is the number you add.