For which value of n will Sₙ − S₍ₙ₋₄₎ = 282, where Sₙ = 1 + 2 + ⋯ + n?

Correct answer: 72

Direct answer: option C, 72. The sum Sₙ contains terms 1 through n, while S₍ₙ₋₄₎ contains terms 1 through n − 4. On subtraction, the common terms cancel, leaving four terms: (n − 3) + (n − 2) + (n − 1) + n. Combining them gives 4n − 6. Set this equal to the given value: 4n − 6 = 282. Adding 6 gives 4n = 288, and dividing by 4 gives n = 72. Option A, 70, gives 274; option B, 71, gives 278; option C gives 282 exactly; option D, 73, gives 286. Thus only C works. A common confusion is to write the remaining terms as n − 4 through n, which would incorrectly leave five terms. Memory cue: subtracting S₍ₙ₋₄₎ leaves the last four terms of Sₙ.