Stared at this for a bit trying to find one clean rule across all nine terms and got nowhere fast.
First, look for a pattern by examining differences, ratios, or alternating subsequences. Since the sequence has 9 terms, consider splitting into odd and even positions, as they often follow separate rules. Then verify the pattern and compute the missing value.
Pro tip: At Optiver, interviewers value clear, logical reasoning over a quick answer. Verbalize your thought process, test multiple hypotheses, and explain why you reject alternatives.
Notice the sequence length (9 terms) and consider splitting into odd and even indexed subsequences. This is a common pattern in number sequences.
List the 1st, 3rd, 5th, 7th, and 9th terms: 2, 3, 5, 8, 16. Look for a pattern: differences are 1, 2, 3, 8; not obvious. Check ratios: 3/2=1.5, 5/3≈1.67, 8/5=1.6, 16/8=2. Alternatively, consider that each term might be the sum of the previous two odd terms? 2+3=5, 3+5=8, 5+8=13 (not 16). So not Fibonacci. Another idea: each term is previous term plus increasing powers? 2+1=3, 3+2=5, 5+3=8, 8+8=16? That would be adding 1,2,3,8—not consistent. Perhaps the odd subsequence is 2,3,5,8,16: each term is the sum of the previous two? 2+3=5, 3+5=8, 5+8=13 (not 16). So no. Maybe it's 2,3,5,8,16: each term is previous term multiplied by something? 2*1.5=3, 3*1.67=5, 5*1.6=8, 8*2=16. Not clear. But note that 16 is 2^4, 8 is 2^3, 5 is not a power of 2. So maybe the odd subsequence is not the key.
List the 2nd, 4th, 6th, and 8th terms: 2, 4, 8, ?. This is a clear geometric progression: each term doubles. So the missing term is 16.
Check if the odd subsequence also follows a pattern: 2, 3, 5, 8, 16. Notice that 2+3=5, 3+5=8, but 5+8=13, not 16. However, 8+8=16? That would require the previous odd term to be 8, which it is. So maybe the rule is: each odd term is the sum of the previous two odd terms, except the last one? That seems inconsistent. Alternatively, the odd subsequence might be: 2, 3, 5, 8, 16 where each term is the sum of the previous term and the previous even term? Let's see: 3 = 2+1? Not clear. But the even subsequence is perfectly consistent, so the missing number is 16.
The missing number is 16. The pattern is that the sequence alternates between two subsequences: odd positions follow a pattern (possibly Fibonacci-like but with a twist), and even positions follow a geometric progression doubling each time.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.