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What should come in place of the question mark (?) in the given series? 2, 5, 13, 28, 52, 87, ?
130
98
107
135
135
Calculate the differences between consecutive terms, then examine the second level of differences to identify a clear pattern of increasing squares.
Calculate the differences between consecutive terms, then examine the second level of differences to identify a clear pattern of increasing squares.
Calculate first-level differences
Subtract consecutive terms: 5-2 = 3, 13-5 = 8, 28-13 = 15, 52-28 = 24, 87-52 = 35.
Identify the second-level logic
Look at the differences (3, 8, 15, 24, 35): 8-3 = 5, 15-8 = 7, 24-15 = 9, 35-24 = 11. The sequence of second-level differences follows consecutive odd numbers (5, 7, 9, 11).
Calculate the next term
The next second-level difference must be 13. Adding 13 to the last first-level difference (35) gives 48. Finally, add 48 to the last term (87) to get 135.
A: 130 is incorrect because it fails to satisfy the established second-level increment of 13 added to 35. B: 98 is too low as it only accounts for a difference of 11, missing the required progression. C: 107 is incorrect as it results from a pattern error in the second-level calculation.
D is correct because the series follows a pattern where the difference between terms increases by consecutive odd numbers (5, 7, 9, 11, 13), leading to 87 + (35 + 13) = 135.
When first-level differences don't show an obvious pattern, always check the difference of the differences; this often reveals squares, cubes, or arithmetic progressions.
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