Interleaved Number Series: Choosing the Rule That Explains Every Term
A fifth-grade enrichment group is solving a number series: 2, 10, 4, 20, 8, 40, ?. The answer choices are 16, 50, 80, and 120. Jonah says 80: the last term is 40, and the visible pattern includes doubling, so the next term should double 40. Lena says 16: she writes the terms in two rows, odd positions (2, 4, 8, ?) and even positions (10, 20, 40), and notices that each row doubles. Maya says 50: she sees 2 + 8 = 10, 4 + 16 = 20, 8 + 32 = 40, so the next first-row term plus 64 should equal 80, but she cannot complete the rule without inventing terms. The teacher asks the group to test each rule against all six given terms and to prefer the rule that accounts for every term without exceptions.
1. Which answer is best? Test each proposed rule against the full sequence.
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16. Lena’s two-row rule accounts for all six terms: odd positions 2, 4, 8 double to 16; even positions 10, 20, 40 double. The missing seventh term is an odd-position term, so it is 8 × 2 = 16.
Jonah’s single doubling rule fails immediately at 10 → 4: 10 doubled is 20, not 4. It works only if one ignores the alternating structure. Maya’s addition rule is not a rule yet: it uses invented future terms (16, 32, 64) that are not in the given sequence, and it does not generate the given terms in order. A valid series rule must generate every displayed term, not just the last jump. The best answer uses the simplest consistent structure: two interleaved geometric sequences.
2. Jonah says a series can have more than one pattern, so 80 is also 'correct.' Explain the standard used to choose between competing patterns.
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Many finite sequences can be fit by more than one formula, so the test uses a practical standard: the intended rule should be simple, use the given terms, and account for every term without exceptions. Jonah’s rule has an exception at 10 → 4, so it is not a rule for this series; it is a local observation about the last step. If a student can produce an alternative rule that is equally simple and explains all six terms, the item may be ambiguous. Here Lena’s interleaved rule is simpler than any contrived polynomial or invented-term rule. The key judgement is not “Can I invent a pattern?” but “Which pattern is supported by all the evidence?”
3. If the sequence were 2, 10, 4, 20, 8, 40, 16, what would be the next term under the same rule, and how would you check?
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The next term would be 80. After 16, the next position is even: the even row runs 10, 20, 40, so the next even-row term is 80. Check the odd row: 2, 4, 8, 16 also doubles. The sequence alternates odd and even rows, so the eighth term is 80. This confirms the rule continues. If a proposed answer ignored the alternation and doubled 16 to get 32, it would put two odd-row terms in a row and break the even-row progression.