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Find the missing number in the series: 2, 6, 12, 20, 30, ?
40
42
44
36
42
Observe the differences between consecutive terms (4, 6, 8, 10, 12) or apply the formula n squared plus n (n² + n).
Observe the differences between consecutive terms (4, 6, 8, 10, 12) or apply the formula n squared plus n (n² + n).
Analyze consecutive differences
Find the difference between consecutive numbers: 6-2=4, 12-6=6, 20-12=8, 30-20=10. The differences are increasing by 2 each time.
Determine the next term
Following the pattern of even number differences (4, 6, 8, 10), the next difference must be 12. Adding this to the last term: 30 + 12 = 42.
Verification via algebraic formula
The series follows the formula n(n+1) starting from n=1: 1(2)=2, 2(3)=6, 3(4)=12, 4(5)=20, 5(6)=30. The next term for n=6 is 6(7) = 42.
A: 40 is incorrect because it implies an increment of only 10, failing the pattern of increasing even differences. C: 44 is incorrect as it assumes a jump of 14 instead of 12. D: 36 is incorrect because the difference from 30 would only be 6, disrupting the established +2 increment pattern.
B is correct because the series follows an increasing gap pattern of 4, 6, 8, 10, and 12, making 30 + 12 = 42.
For series problems, always check the difference of differences if the first layer does not immediately reveal a clear constant or simple arithmetic pattern.
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