Definition: A recursive formula defines each term of a sequence from the term or terms before it, and it must also state the starting term or terms.
How does a recursive formula work?
Instead of jumping straight to the 50th term, you walk there one step at a time. The rule says how to get the next term from the current one, and the starting value tells you where the walk begins.
What does it look like?
For an arithmetic sequence: a1 = first value, an = an-1 + d. For a geometric sequence: a1 = first value, an = r · an-1.
Example. Let a1 = 6 and an = 2an-1 – 3. Then a2 = 2(6) – 3 = 9, a3 = 2(9) – 3 = 15, and a4 = 2(15) – 3 = 27. The list is 6, 9, 15, 27.
What is the typical error?
Forgetting the starting term. Without it, many different sequences fit the same rule. Another error is plugging n into the rule instead of the previous term. Recursive rules are slow for far-away terms, which is why an explicit formula is better when you need the 200th term. Recursion shines when the pattern itself is described by the step, as in loans or populations.
Where to learn it
- Arithmetic and Geometric Sequences as Functions – lesson
- Building Functions from Real Situations – lesson