In the number series 6, 15, 34, 63, 102, the next term is _______. – UGC NET Question

The first-level differences are 9, 19, 29 and 39. These differences themselves form an arithmetic sequence with common difference 10. To preserve this pattern, the next difference must be 49. Adding 49 to 102 gives 151 as the next term. Thus 151 is the only option that keeps the second-level pattern perfectly intact.

Option A:
Option A corresponds to adding 49 to 102, giving 151 and extending the differences to 9, 19, 29, 39, 49. This preserves the constant increase of 10 between successive differences. Because both the original series and the sequence of gaps remain consistent, 151 is the correct continuation.

Option B:
Option B would make the next difference 47, so the differences become 9, 19, 29, 39, 47. The jump from 39 to 47 is 8, not 10, so the second-level pattern breaks. Hence 149 does not follow the intended rule of steadily increasing differences by 10.

Option C:
Option C would require a difference of 51 from 102, giving differences 9, 19, 29, 39, 51. Here the increment from 39 to 51 is 12, which again violates the constant second-level difference of 10. Therefore 153 cannot be accepted as the next term.

Option D:
Option D implies a difference of 45 from 102, so the last increment in the difference sequence is only 6. This contradicts the observed +10 progression that generates 9, 19, 29 and 39. As a result, 147 is not a valid continuation of the series.