Explicit formula

Definition

An explicit formula directly defines the nth term of a sequence as a function of n. Unlike recursive formulas, it does not require the computation of previous terms.

5 Must Know Facts For Your Next Test

1. An explicit formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and d is the common difference.
2. An explicit formula for a geometric sequence is $a_n = a_1 \cdot r^{(n-1)}$, where $a_1$ is the first term and r is the common ratio.
3. Explicit formulas allow you to find any term in a sequence without having to know or calculate previous terms.
4. In sequences, explicit formulas are often easier to work with when dealing with large values of n.
5. Explicit formulas can be derived from recursive formulas but typically require algebraic manipulation.

Review Questions

• What is the explicit formula for an arithmetic sequence with first term 3 and common difference 5?
• How do you derive an explicit formula from a given recursive formula?
• Why might using an explicit formula be more efficient than using a recursive one?

Related terms

Arithmetic Sequence: A sequence in which each term after the first is obtained by adding a constant difference to the preceding term.

Geometric Sequence: A sequence in which each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number called the common ratio.

Recursive Formula: $\text{A formula that requires knowing one or more previous terms in order to find any subsequent term}$.