Q3 — Triangular number membership (3 marks)

Question: Show that 91 is a triangular number and state its position in the sequence.

Mark scheme (3 marks):

• Use $T_n = \tfrac{1}{2}n(n+1) = 91$ giving $n^2 + n - 182 = 0$ (M1).
• Solve: $(n-13)(n+14) = 0$ so $n = 13$ (rejecting $n = -14$) (M1).
• Conclude: 91 is the 13th triangular number (A1).
• Alternative method: list $T_{12} = 78$, $T_{13} = 91$ (full 3 marks if shown systematically).