In the AP (53,46,39,\ldots), which is the first term less than (-100)?

Correct answer: (-101)

The direct answer is option B, -101. The sequence is an arithmetic progression: 53, 46, 39, ... Each term decreases by 7, so the common difference is d=-7. We need the first term that is less than -100, not merely equal to or close to it. Continue the sequence near the target: after 53 and repeated subtraction of 7, the terms are ...,-87,-94,-101,-108. The term -94 is not less than -100 because -94 is greater than -100. The next term is -101, and -101<-100, so it is the first qualifying term. Option A, -94, is wrong because it has not crossed below -100. Option B, -101, is correct because it is below -100 and comes immediately after -94. Option C, -108, is also less than -100, but it is not the first such term; -101 occurs earlier. Option D, -115, is likewise below -100 but occurs even later. Algebraically, the nth term is a_n=53+(n-1)(-7)=60-7n. Solving 60-7n<-100 gives -7n<-160; reversing the sign when dividing by -7 gives n>160/7≈22.86, so the first integer n is 23. Then a_23=60-161=-101. The key memory point is that with negative numbers, -101 is smaller than -100, while -94 is larger.