I stared at this way too long looking for an arithmetic or geometric pattern before accepting it was something nonlinear.
First, look for patterns in differences, ratios, or combinations of previous terms. Test simple recurrence relations like a_n = a_{n-1} * k + c or a_n = a_{n-1} * a_{n-2} + c. Once a rule fits all given terms, apply it to find the missing term and verify consistency.
Pro tip: In interviews, verbalize your thought process and consider multiple hypotheses before settling on one. If a rule seems too complex, simplify by checking if operations alternate or involve fractions as a clue.
Write down the terms: 2, 1, 2, 5, 13, 73/2. Notice the fraction suggests division or multiplication by fractions may be involved.
Calculate successive differences: -1, +1, +3, +8, +47/2. Ratios: 1/2, 2, 5/2, 13/5, 73/26. Look for patterns in these derived sequences.
Try relations like a_n = a_{n-1} * a_{n-2} + c or a_n = a_{n-1} * k + a_{n-2}. For example, check if a_n = a_{n-1} * a_{n-2} + 1 works: 2*1+1=3 (no), but 1*2+1=3 (no). Try a_n = a_{n-1} * (n-1) + a_{n-2}? Systematically test simple operations.
Once a rule fits all terms, apply it to the last term to find the missing term. For instance, if the rule is a_n = a_{n-1} * a_{n-2} + a_{n-3}, verify with given terms and compute.
Check that the computed term satisfies the rule and explain the pattern clearly, mentioning any assumptions or alternative rules considered.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.