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Complete the series: 3, 6, 12, 21, 33, ?
45
48
51
54
48
Find the differences between consecutive terms to reveal an arithmetic progression of differences (3, 6, 9, 12, 15).
Find the differences between consecutive terms to reveal an arithmetic progression of differences (3, 6, 9, 12, 15).
Calculate First-Level Differences
Find the difference between each consecutive pair of numbers: 6 - 3 = 3, 12 - 6 = 6, 21 - 12 = 9, and 33 - 21 = 12.
Identify the Difference Pattern
The differences (3, 6, 9, 12) form an arithmetic progression where each difference increases by 3.
Compute the Missing Term
Following the pattern, the next difference must be 12 + 3 = 15. Adding this to the last term gives 33 + 15 = 48.
A: 45 is incorrect because it results from an incorrect addition step. C: 51 is incorrect because it would require a difference of 18. D: 54 is incorrect because it would require a difference of 21.
B is correct because the differences between consecutive terms increase by 3 sequentially (3, 6, 9, 12), meaning the next increment is 15, yielding 33 + 15 = 48.
When first-level differences do not yield a constant value, compute the second-level differences to easily spot polynomial or arithmetic progressions.
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