The value of Sₙ − S₍ₙ₋₁₎ + S₍ₙ₋₄₎ − S₍ₙ₋₅₎ is 150. What is n, where Sₖ = 1 + 2 + ⋯ + k?

Correct answer: 77

Direct answer: option C, 77. The basic identity is Sₖ − S₍ₖ₋₁₎ = k, because all terms up to k − 1 cancel and only k remains. Apply it to the first pair: Sₙ − S₍ₙ₋₁₎ = n. Apply it to the second pair with k = n − 4: S₍ₙ₋₄₎ − S₍ₙ₋₅₎ = n − 4. Therefore the whole expression is n + (n − 4) = 2n − 4. Equating this with 150 gives 2n − 4 = 150, so 2n = 154 and n = 77. Option A gives 146, option B gives 148, option C gives 150, and option D gives 152. Thus C is the only valid choice. Memory cue: pair consecutive S terms first; each pair collapses to the newly added endpoint.