Stared at this for way too long looking for an additive pattern before realizing you need to look at pairs.
First, look for patterns in the differences or ratios between consecutive terms. Notice that the sequence alternates between multiplying by a fraction and adding a constant: 12 * (7/3) = 28, 28 + 8 = 36, 36 * (7/3) = 84, 84 + 4 = 88, 88 * (7/3) = 168? Wait, 88 * (7/3) is not 168. Actually, 88 * (21/11) = 168, but that's not consistent. Alternatively, observe that the sequence can be split into two interleaved subsequences: odd positions (12, 36, 88, ?) and even positions (28, 84, 168). The even positions follow a clear pattern: 28 * 3 = 84, 84 * 2 = 168. The odd positions: 12 * 3 = 36, 36 * (22/9) = 88? That's not clean. Another approach: look at the multipliers between terms: 28/12 = 7/3, 36/28 = 9/7, 84/36 = 7/3, 88/84 = 22/21, 168/88 = 21/11. Not obvious. Perhaps the sequence is formed by multiplying by 2 and adding/subtracting? 12*2+4=28, 28+8=36, 36*2+12=84, 84+4=88, 88*2-8=168? Not consistent. Let's try: 12*2+4=28, 28+8=36, 36*2+12=84, 84+4=88, 88*2-8=168? No. Maybe the pattern is: multiply by 2 and add 4, then add 8, then multiply by 2 and add 12, then add 4, then multiply by 2 and subtract 8? That seems arbitrary. Another idea: the sequence might be based on multiplying by 2 and adding 4, then adding 8, then multiplying by 2 and adding 12, then adding 4, then multiplying by 2 and adding 8? 88*2+8=184, but 168 is 88*2-8. Hmm. Let's check if the sequence is: 12*2+4=28, 28+8=36, 36*2+12=84, 84+4=88, 88*2-8=168? That gives 168, but then next? If pattern of multipliers: *2+4, +8, *2+12, +4, *2-8, then next might be +8? 168+8=176. But is that consistent? The operations: multiply by 2 and add 4, then add 8, then multiply by 2 and add 12, then add 4, then multiply by 2 and subtract 8. The added numbers: 4, 8, 12, 4, -8? Not a clear pattern. Alternatively, maybe the sequence is: 12*2+4=28, 28+8=36, 36*2+12=84, 84+4=88, 88*2-8=168, then 168+16=184? That would be adding 16. But the additions: 4,8,12,4,-8,16? Not clear. Another common pattern: multiply by 2 and add 4, then add 8, then multiply by 2 and add 12, then add 4, then multiply by 2 and add 8? That would be 88*2+8=184, not 168. So maybe the pattern is: multiply by 2 and add 4, then add 8, then multiply by 2 and add 12, then add 4, then multiply by 2 and subtract 8? That gives 168. Then next might be add 16? 168+16=184. But is there a pattern in the added numbers? 4,8,12,4,-8,16? Not obvious. Perhaps the sequence is: 12*2+4=28, 28+8=36, 36*2+12=84, 84+4=88, 88*2-8=168, then 168+16=184? That would be adding 16. But why -8 then +16? Maybe the pattern is: multiply by 2 and add 4, then add 8, then multiply by 2 and add 12, then add 4, then multiply by 2 and add 8? That would be 184, but the given term is 168, so that doesn't work. So the pattern must be something else. Let's try: 12*2+4=28, 28+8=36, 36*2+12=84, 84+4=88, 88*2-8=168. The added numbers: 4,8,12,4,-8. The multipliers: 2,1,2,1,2. The added numbers might be: 4,8,12,4,-8,? Not clear. Another idea: the sequence might be formed by multiplying by 2 and adding 4, then adding 8, then multiplying by 2 and adding 12, then adding 4, then multiplying by 2 and adding 8? But that gives 184, not 168. So maybe the pattern is: multiply by 2 and add 4, then add 8, then multiply by 2 and add 12, then add 4, then multiply by 2 and subtract 8, then add 16? That would be 184. But is there a pattern in the added numbers? 4,8,12,4,-8,16? Not obvious. Perhaps the sequence is: 12*2+4=28, 28+8=36, 36*2+12=84, 84+4=88, 88*2-8=168, then 168+16=184? That would be adding 16. But why -8 then +16? Maybe the pattern is: multiply by 2 and add 4, then add 8, then multiply by 2 and add 12, then add 4, then multiply by 2 and add 8? That would be 184, but the given term is 168, so that doesn't work. So the pattern must be something else. Let's try: 12*2+4=28, 28+8=36, 36*2+12=84, 84+4=88, 88*2-8=168. The added numbers: 4,8,12,4,-8. The multipliers: 2,1,2,1,2. The added numbers might be: 4,8,12,4,-8,? Not clear. Another idea: the sequence might be formed by multiplying by 2 and adding 4, then adding 8, then multiplying by 2 and adding 12, then adding 4, then multiplying by 2 and adding 8? But that gives 184, not 168. So maybe the pattern is: multiply by 2 and add 4, then add 8, then multiply by 2 and add 12, then add 4, then multiply by 2 and subtract 8, then add 16? That would be 184. But is there a pattern in the added numbers? 4,8,12,4,-8,16? Not obvious. Perhaps the
Pro tip: Take a moment to organize your thoughts before answering.
Clarify what is being asked.
Organize your key points.
Present your answer clearly.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
The fractions at the start threw me off completely.
Start by examining differences between consecutive terms to identify a pattern. If differences don't yield a clear rule, try ratios or alternating operations. Once a rule is hypothesized, test it on all given terms before predicting the next term.
Pro tip: In interviews, verbalize your thought process and consider multiple hypotheses; even if you don't find the intended rule, demonstrating systematic exploration and adaptability is highly valued.
Calculate the differences between consecutive terms: 1 - 4/3 = -1/3, 5 - 1 = 4, 8 - 5 = 3, 23 - 8 = 15, 47 - 23 = 24. Look for patterns in these differences.
If differences are not obvious, compute ratios or second differences. For example, ratios: 1/(4/3)=3/4, 5/1=5, 8/5=1.6, 23/8=2.875, 47/23≈2.043. Alternatively, look at operations like multiply by n and add/subtract.
Notice that from the second term onward, the sequence might follow: multiply by 1 and add 0? Actually, test: 1*2+3=5, 5*2-2=8, 8*3-1=23, 23*2+1=47. This suggests alternating multipliers and adjustments.
Apply the identified rule to the last term to find the next term. For instance, if the pattern is multiply by 2 and add increasing numbers: 1*2+3=5, 5*2-2=8, 8*3-1=23, 23*2+1=47, then next might be 47*3+? = ?
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This one looked deceptively simple and I rushed it.
Start by examining the sequence for patterns involving operations on previous terms, such as alternating operations or combinations of addition and multiplication. Test hypotheses like a_n = a_{n-1} * a_{n-2} + something, or alternating between addition and multiplication. Once the rule is identified, apply it to find the next term.
Pro tip: Verbalize your thought process clearly, showing how you test and discard hypotheses, because interviewers at Optiver value logical reasoning and adaptability over just the final answer.
Check differences, ratios, and common sequences (arithmetic, geometric, Fibonacci-like) to see if any obvious pattern emerges.
Test if each term is derived from the previous one or two terms using operations like addition, multiplication, or exponentiation.
Try rules that alternate between operations (e.g., +, *, +, *) or combine two previous terms (e.g., a_n = a_{n-1} * a_{n-2} + c).
Ensure the hypothesized rule holds for every consecutive pair or triple in the sequence to avoid coincidental matches.
Use the confirmed rule to compute the missing term and double-check the calculation.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Numerators and denominators each follow their own separate rule.
Treat the sequence as two interleaved subsequences: one for numerators and one for denominators. Analyze each subsequence separately to identify patterns, then combine to find the next term.
Pro tip: When faced with a fraction sequence, always check if numerators and denominators follow independent patterns. Also, consider that the sequence might be designed to test your ability to handle ambiguity and think aloud, so verbalize your reasoning clearly.
List the numerators: 1, 2, 3, 3, 5, 4. List the denominators: 5, 3, 11, 6, 17, 9.
Look for patterns in the numerator sequence. Check differences, alternating patterns, or relationships to position index.
Look for patterns in the denominator sequence. Check differences, alternating patterns, or relationships to position index.
Use the identified patterns to determine the next numerator and denominator, then form the fraction.
Check if the predicted term fits any other plausible pattern to ensure consistency.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, look for simple arithmetic patterns like differences or ratios, but be prepared for a more complex rule such as alternating operations or modular arithmetic. Since the sequence jumps around, consider grouping terms or using a cycle, and test your rule against all given terms before predicting the next.
Pro tip: In interviews, it's often more important to clearly explain your reasoning and how you'd test hypotheses than to find the 'correct' answer immediately. If you get stuck, verbalize alternative patterns you're considering and why they fit or don't fit.
Compute differences and ratios between consecutive terms to see if there's a simple arithmetic or geometric progression.
Examine odd and even positions separately, or look for a repeating cycle of operations (e.g., +7, +10, -18, +6, +10, ...).
Check if terms relate to digits (e.g., sum of digits, product) or if the sequence follows a rule modulo some number.
For any rule you hypothesize, verify it against all given terms. If it fails, discard and try another.
Once a consistent rule is found, use it to compute the next term and briefly explain your reasoning.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Grouping in sets of four seemed to be the move.
First, look for patterns in the differences or ratios between consecutive terms, and consider grouping terms into blocks. Notice that the sequence can be split into groups of four where each group follows a similar transformation. Identify the rule within each group and apply it to find the next term.
Pro tip: In interviews, clearly verbalize your thought process and test multiple hypotheses quickly; even if you don't find the answer immediately, demonstrating structured reasoning is highly valued.
Calculate the differences between consecutive terms: -1, +2, +40, -45, -1, +2, ?. Notice the pattern of -1, +2 repeats, but the large jump suggests a multiplicative step.
Group the sequence into blocks of four: (19, 18, 20, 60) and (15, 14, 16, ?). Observe that the first three numbers in each block decrease by 1, then increase by 2, and the fourth number is the third multiplied by 3.
In the first block: 19 - 1 = 18, 18 + 2 = 20, 20 * 3 = 60. In the second block: 15 - 1 = 14, 14 + 2 = 16, so 16 * 3 = 48.
The next term is the fourth number of the second block, which is 48.
Check that the pattern holds for all given terms and that the next term logically follows the established rule.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by examining the differences and ratios between consecutive terms to identify a pattern. Test common sequence types (arithmetic, geometric, recursive with operations) and consider that the rule may involve alternating operations or increasing multipliers. Once a rule is hypothesized, verify it against all given terms before computing the next term.
Pro tip: In interviews, clearly verbalize your thought process and test multiple hypotheses; even if you don't find the intended rule immediately, demonstrating systematic reasoning and adaptability is highly valued. Also, consider that sequences can have multiple valid rules, so be prepared to justify your answer.
Calculate the differences and ratios between consecutive terms to see if there is a simple arithmetic or geometric pattern. For 4,2,6,6,30,150: differences are -2,4,0,24,120; ratios are 0.5,3,1,5,5.
Check if operations alternate (e.g., divide by 2, multiply by 3, add 0, multiply by 5, multiply by 5) or if each term depends on previous terms with changing multipliers. Notice that 4/2=2, 2*3=6, 6*1=6, 6*5=30, 30*5=150.
The multipliers appear to be 0.5, 3, 1, 5, 5. This could be a sequence itself: 0.5, 3, 1, 5, 5. Look for a pattern in these multipliers (e.g., they might be derived from previous terms or follow a separate rule).
If the multipliers are 0.5, 3, 1, 5, 5, the next multiplier might be 7 (if odd numbers increasing: 1,3,5,7) but 0.5 breaks that. Alternatively, consider that the multipliers could be the digits of the previous term? 4/2=2, 2*3=6, 6*1=6, 6*5=30, 30*5=150. The multipliers 2,3,1,5,5 might be the digits of the terms? 4,2,6,6,30,150: digits: 4,2,6,6,3,0,1,5,0. Not obvious.
Another common pattern: a_n = a_{n-1} * a_{n-2} / something? 4*2=8, not 6. Or a_n = a_{n-1} + a_{n-2} * something? 4+2=6 (third term), 2+6=8 not 6. Or a_n = a_{n-1} * (n-1) + something? Check: 4*0.5=2, 2*3=6, 6*1=6, 6*5=30, 30*5=150. The multipliers 0.5,3,1,5,5 could be 1/2, 3, 1, 5, 5. Maybe they are the prime numbers? 2,3,5,7,11 but with 1 and 5 repeated? Not clear.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Classic recurrence relation type thing, each term is roughly double the previous plus something.
First, examine the differences between consecutive terms to see if they follow a simple pattern. If not, look for a recurrence relation such as each term being a linear combination of previous terms. Then, verify the rule and compute the next term.
Pro tip: In interviews, clearly verbalize your thought process and consider multiple hypotheses before settling on one. Mentioning that you'd verify with more terms shows rigor.
Calculate the differences between consecutive terms: 2, 4, 10, 24, 58. Check if these follow a pattern (e.g., doubling, primes, etc.).
Examine the ratios of consecutive terms: 3, 2.33, 2.43, 2.41, 2.41. They seem to approach a constant, suggesting a recurrence relation.
Try common recurrences like a_n = 2*a_{n-1} + a_{n-2} or a_n = 2*a_{n-1} + something. For example, 3 = 2*1 + 1, 7 = 2*3 + 1, 17 = 2*7 + 3, 41 = 2*17 + 7, 99 = 2*41 + 17. This fits a_n = 2*a_{n-1} + a_{n-2}.
Check that the recurrence holds for all given terms. For n=3: 2*3+1=7, n=4: 2*7+3=17, n=5: 2*17+7=41, n=6: 2*41+17=99. Yes, it works.
Apply the recurrence to find the next term: a_7 = 2*99 + 41 = 198 + 41 = 239.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.