If Sₙ − Sₙ₋₃ = 114, what is the value of n?

Correct answer: 39

Direct answer: n = 39, so option C is correct. Write the sums explicitly: Sₙ = 1 + 2 + ... + (n − 3) + (n − 2) + (n − 1) + n, while Sₙ₋₃ = 1 + 2 + ... + (n − 3). When the second is subtracted from the first, all common terms cancel, leaving (n − 2) + (n − 1) + n. Add these: n − 2 + n − 1 + n = 3n − 3. The condition becomes 3n − 3 = 114. Add 3: 3n = 117; divide by 3: n = 39. Check: the final three terms are 37, 38, and 39, and their sum is 114. Thus C is correct. For comparison, n = 36 gives 105, n = 38 gives 111, and n = 42 gives 123. Common confusion: subtracting the two sums leaves three terms, not n − 3 or merely the difference of the subscripts.