The $n$th term formula lets you find any term in a sequence without listing them all, and test whether a given number is in the sequence.
Linear sequences (arithmetic):
Find the common difference $d$ — this is the coefficient of $n$. Adjust with a constant to match the first term.
Example: 5, 8, 11, 14, …
- Difference = 3, so starts with $3n$
- When $n = 1$: $3(1) = 3$, but the first term is 5, so add 2
- $n$th term: $3n + 2$
Check: $n = 4 \Rightarrow 3(4) + 2 = 14$ ✓
Is 50 in the sequence? $$3n + 2 = 50 \Rightarrow n = 16 \quad \checkmark \text{ (whole number, so yes)}$$
Is 51 in the sequence? $$3n + 2 = 51 \Rightarrow n = \frac{49}{3} \quad \text{(not a whole number, so no)}$$
Descending sequences: the difference is negative. $$20, 17, 14, 11, \ldots \Rightarrow -3n + 23$$
Common error: writing $3n$ when the difference is 3 but not adjusting for the starting value.