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Find the next term: 2, 6, 12, 20, 30, ?
40
42
44
46
42
Observe the differences between consecutive terms: +4, +6, +8, +10. The gap increases by 2 each time, meaning the next difference is +12.
Observe the differences between consecutive terms: +4, +6, +8, +10. The gap increases by 2 each time, meaning the next difference is +12.
Analyze consecutive differences
Calculate the difference between adjacent terms: 6-2=4, 12-6=6, 20-12=8, 30-20=10. The differences are 4, 6, 8, 10, which forms an arithmetic progression with a common difference of 2.
Calculate the next term
Following the pattern, the next difference must be 10 + 2 = 12. Therefore, the next term is 30 + 12 = 42.
Alternative approach (Square pattern)
The terms can be expressed as n² + n: (1²⁺¹)=2, (2²⁺²)=6, (3²⁺³)=12, (4²⁺⁴)=20, (5²⁺⁵)=30. Thus, (6²⁺⁶)=36+6=42.
A: 40 is incorrect as it results from adding only 10 instead of 12; C: 44 is incorrect as it results from adding 14; D: 46 is incorrect as it results from adding 16.
B is correct because the series follows a pattern of adding consecutive even numbers (+4, +6, +8, +10, +12), resulting in 30+12=42.
For any series, if the first difference is inconsistent, always check the 'second difference' (the difference of the differences) to find a constant pattern.
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