Nth term of the sequence

Definition

The nth term of a sequence is a formula that allows you to find the value of any term in the sequence based on its position number, n. It generalizes the pattern within the sequence for all terms.

5 Must Know Facts For Your Next Test

1. The nth term formula often involves arithmetic or geometric expressions.
2. In an arithmetic sequence, the nth term is given by $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.
3. In a geometric sequence, the nth term is given by $a_n = a_1 \cdot r^{(n-1)}$, where $a_1$ is the first term and $r$ is the common ratio.
4. Finding the nth term helps in predicting future values without listing all preceding terms.
5. Recursive sequences require previous terms to determine subsequent ones, unlike explicit formulas which directly use n.

Review Questions

• What is the formula for finding the nth term of an arithmetic sequence?
• How does one derive the nth term for a geometric sequence?
• Why might it be useful to have an explicit formula for determining any term in a sequence?

Related terms

Arithmetic Sequence: A sequence in which each term after the first is found by adding a constant to its predecessor.

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

Recursive Formula: \nA formula that requires one or more of the previous terms to find subsequent terms in a sequence.