From the following which statement is correct in regards to the given series?…

Concept

A "second-difference" number series does not increase by a constant gap; instead, the gaps between consecutive terms themselves grow according to a further rule. To test any term written in brackets, first derive that governing rule using only the terms whose values are given as fixed anchors, then apply the derived rule forward -- never accept a bracketed term's own arithmetic at face value, since any inserted number will always look locally consistent with its immediate neighbours.

Step-by-step application

1. Write out the series with positions: T1 = 49, T2 = 66, T3 = 89, T4 = (124), T5 = 165, T6 = 218, T7 = (278).

2. Using only the fixed (non-bracketed) terms, find the differences: T2 minus T1 = 17, T3 minus T2 = 23, and T6 minus T5 = 53.

3. Find the second difference from the fixed data: 23 minus 17 = 6, which suggests the gaps grow by an alternating +6, +12, +6, +12 pattern.

4. Apply the pattern forward from T3: the next gap is 23 + 12 = 35, so the predicted T4 = 89 + 35 = 124 -- this matches the bracketed value.

5. Continue: the next gap is 35 + 6 = 41, so the predicted T5 = 124 + 41 = 165 -- this matches the independently fixed T5, confirming the pattern is correct.

6. Continue: the next gap is 41 + 12 = 53, so the predicted T6 = 165 + 53 = 218 -- this again matches the independently fixed T6, a second confirmation.

7. Continue one more step: the next gap is 53 + 6 = 59, so the predicted T7 = 218 + 59 = 277 -- this does NOT match the bracketed value of 278.

Cross-check

Two consecutive terms, T5 and T6, were predicted purely from the reconstructed alternating +6/+12 rule after inserting T4 = 124, and both landed exactly on the values already fixed in the series -- so the rule itself is verified independently, not merely assumed to fit. The one place the rule and the printed series disagree is the very last entry, which isolates the error to that single value rather than to the rule or to the earlier bracket.

Result

The first bracketed number, 124, satisfies the series rule and is correct. The second bracketed number, 278, does not satisfy the rule (the rule gives 277) and is incorrect. So the statement that matches is: the first bracketed number is correct and the second bracketed number is incorrect.