The nth Term

To find the 100th term of a sequence, listing 99 terms first would take a while. A rule gets there in one step. It is an expression in n, the term's position: the first term has n = 1, the second n = 2, and so on.

Take the rule $4n+1$. The first term is 4 · 1 + 1, which is 5. The second is 4 · 2 + 1, which is 9, then 13, 17, and so on: a common difference of 4. For the 10th term, substitute n = 10 straight away: $t = 4\cdot 10+1$, so $t = 41$.

Notice that n is the position, not the term. The 10th term is 41: n is 10 and the term is 41. Mixing the two up is the usual slip, so say it to yourself: n counts places, the rule gives the number in that place.