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Identify the next term in the series: 2, 6, 12, 20, 30, ?
42
40
38
44
42
Recognize the series as consecutive products of integers, i.e., n x (n + 1) where n starts from 1.
Recognize the series as consecutive products of integers, i.e., n x (n + 1) where n starts from 1.
Analyze the differences between consecutive terms
Calculate the difference between adjacent terms: 6 - 2 = 4, 12 - 6 = 6, 20 - 12 = 8, 30 - 20 = 10. The differences form an arithmetic progression increasing by 2: 4, 6, 8, 10.
Determine the next difference and calculate the missing term
Following the pattern of differences (4, 6, 8, 10), the next difference must be 12. Add this difference to the last term: 30 + 12 = 42. Alternatively, using the formula n x (n + 1): 1x2=2, 2x3=6, 3x4=12, 4x5=20, 5x6=30, so the next is 6x7=42.
B: 40 is incorrect because adding 10 twice or miscalculating the difference sequence leads here. C: 38 is incorrect as it assumes a constant difference or incorrect growth. D: 44 is incorrect as it results from adding 14 instead of 12.
A is correct because the series follows the pattern of differences increasing by 2 (4, 6, 8, 10, 12), giving 30 + 12 = 42.
Always check both the differences of differences (second-order differences) and product patterns like n(n+1) when facing polynomial number series.
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