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In the sequence 2, 6, 12, 20, 30,?, the next number is:
40
42
44
46
42
Identify the difference between consecutive terms; notice that the difference increases by 2 each time (4, 6, 8, 10, 12).
Identify the difference between consecutive terms; notice that the difference increases by 2 each time (4, 6, 8, 10, 12).
Analyze consecutive differences
Calculate the difference between consecutive numbers: 6-2=4, 12-6=6, 20-12=8, 30-20=10. The pattern of differences is 4, 6, 8, 10, which follows an arithmetic progression with a common difference of 2.
Predict the next difference and term
The next difference in the sequence must be 10 + 2 = 12. Therefore, the next term is 30 + 12 = 42.
Alternative Square-based approach
The sequence follows the form n┬▓ + n: 1┬▓тБ║┬╣=2, 2┬▓тБ║┬▓=6, 3┬▓тБ║┬│=12, 4┬▓тБ║тБ┤=20, 5┬▓тБ║тБ╡=30, 6┬▓тБ║тБ╢=42.
A: 40 is incorrect because it assumes a constant difference or wrong increment pattern. C: 44 is incorrect as it skips the next logical term in the n┬▓тБ║тБ┐ sequence. D: 46 is incorrect as it does not follow the consistent +2 progression in differences.
B is correct because the sequence follows the pattern of adding increasing even numbers (4, 6, 8, 10, 12) or the algebraic formula n squared plus n, resulting in 42.
For number series, if differences are not constant, always calculate the 'second difference' (difference of the differences) to quickly reveal underlying arithmetic patterns.