If in an AP (a_2=2x+1), (a_5=5x-2), and (a_8=8x-5), what is (a_{11})?

Correct answer: (11x-8)

Direct answer: Option A, \(11x-8\). In an AP, terms equally far apart have differences proportional to the number of steps. The given terms are at positions 2, 5, and 8, so each pair is separated by three positions. Calculate the difference from the fifth to the eighth term: \(a_8-a_5=(8x-5)-(5x-2)=3x-3\). The 11th term is also three positions after the eighth term, so the same difference must be added: \(a_{11}=a_8+(3x-3)=(8x-5)+(3x-3)=11x-8\). Option A is correct. Option B, \(11x-6\), adds only 2 instead of the required difference and does not follow the given AP. Option C, \(10x-7\), has the wrong coefficient and constant after the correct addition. Option D, \(12x-9\), adds too much and is not the next three-step AP value. Another check is that the expressions change by \(3x-3\): from \(a_2\) to \(a_5\), from \(a_5\) to \(a_8\), and from \(a_8\) to \(a_{11}\). Exam cue: equal gaps in position mean equal changes in an AP.