Trace a Recursive Sequence
A recursive sequence is defined by a starting value and a rule that gives each new term from earlier term(s). To trace it, you build the sequence step by step instead of trying to jump to the end.
Method
- Find the starting value. Identify the first term given by the definition.
- Apply the rule exactly. Use the previous term, or previous terms, as required by the recurrence.
- Compute one term at a time. Write each new value before moving to the next step.
- Continue until you reach the requested term. Keep track of the order so you do not skip any term.
- Simplify the final answer. Reduce fractions, combine integers, or perform any arithmetic needed.
Helpful checks
- Make sure each term uses only the information allowed by the rule.
- Check that your values follow the pattern created by the recurrence.
- If the sequence changes direction or grows quickly, recompute carefully to avoid sign or arithmetic errors.
Tracing recursive sequences is mostly careful bookkeeping: list the terms in order, substitute each value correctly, and verify that the final result is simplified.