A recursive formula requires at least one initial condition to specify the first term of the sequence.
The general form for an arithmetic sequence's recursive formula is $a_{n} = a_{n-1} + d$, where $d$ is the common difference.
For a geometric sequence, the recursive formula is $a_{n} = a_{n-1} \cdot r$, where $r$ is the common ratio.
Recursive formulas can be used to model real-world problems such as population growth and financial calculations.
To find a specific term in a sequence defined by a recursive formula, you must know all prior terms up to that point.
Review Questions
What information do you need to start using a recursive formula?
How does the recursive formula differ for arithmetic and geometric sequences?
Can you provide an example of how to use a recursive formula to find the third term in an arithmetic sequence with $a_1=2$ and $d=3$?
Related terms
Arithmetic Sequence: A sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term.
Geometric Sequence: A sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.
Initial Condition: The starting value or values needed to begin generating terms from a recursive formula.