A sequence defined by an explicit formula or by a recurrence relation

Explicit formula

A sequence is defined by an explicit formula when u_n can be calculated directly in terms of n, without knowing the preceding terms. This is the most practical way of defining a sequence to obtain a term far down the sequence.

Example: $u_n = n^2 - 3$. We can immediately calculate $u_5 = 5^2 - 3 = 22$, without going through $u_0$, $u_1$, $u_2$, $u_3$ or $u_4$.

Recurrence relation

A sequence is defined by a recurrence relation when the first term (e.g. u₀) is given, followed by a rule for calculating u_(n+1) from u_n: u_(n+1) = f(u_n).

Example: u_0 = 2 and u_(n+1) = u_n + 3 for all n. We then calculate step by step: u_1 = u_0 + 3 = 5, u_2 = u_1 + 3 = 8, u_3 = u_2 + 3 = 11.

Comparative table

Type	Advantage	Disadvantage
Explicit formula	Direct calculation of any term	Not always easy to find
Recurrence relation	Easy to construct step by step	Impossible to jump directly to u_100 without calculating all the terms first

Common pitfall

With a sequence defined by a recurrence relation, we cannot calculate u₅₀ simply by substituting n with 50 in the recurrence formula: we must either calculate all the terms one by one, or prove an equivalent explicit formula (which we will learn about for arithmetic and geometric sequences). Also, take care to check whether the relation gives u_(n+1) in terms of u_n, or u_n in terms of u_(n-1): the shift in index changes the calculation.