I stared at this for a bit and my first instinct was Fibonacci-style addition, which was wrong.
First, look for patterns by examining differences, ratios, and combinations of previous terms. Test simple linear recurrences with constant coefficients, and if needed, consider recurrences with varying coefficients or nonlinear terms. Once a rule is found, verify it against all given terms, then extend the sequence and express the recurrence relation clearly.
Pro tip: At Optiver, interviewers value not just the answer but your systematic approach and ability to handle ambiguity. Verbalize your thought process, test multiple hypotheses, and be ready to discuss why you chose a particular rule over others.
Calculate differences, ratios, and second differences between consecutive terms to identify simple arithmetic or geometric patterns.
Assume a recurrence of the form a_n = p*a_{n-1} + q*a_{n-2} + r and solve for constants using the given terms.
If linear recurrences fail, explore recurrences involving operations like multiplication, exponentiation, or coefficients that depend on n.
Once a rule is hypothesized, check it against all provided terms. Then use it to compute the next few terms.
Express the rule formally as a recurrence relation, specifying initial conditions and the formula for subsequent terms.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.