Q7 — Mixed identification and proof (5 marks)

Question: Sequence P: 1, 4, 9, 16, 25, … and Sequence Q: 1, 3, 6, 10, 15, … (a) State the type of each sequence and write down a formula for the nth term of each. (3 marks) (b) Prove algebraically that the sum of two consecutive terms of Sequence Q is always a term of Sequence P. (2 marks)

Mark scheme (5 marks):

• (a) P is the square numbers, $P_n = n^2$ (B1). Q is the triangular numbers, $Q_n = \tfrac{1}{2}n(n+1)$ (B1). Stating both types correctly (B1).
• (b) $Q_n + Q_{n+1} = \tfrac{1}{2}n(n+1) + \tfrac{1}{2}(n+1)(n+2)$ (M1).
• $= \tfrac{1}{2}(n+1)[n + (n+2)] = \tfrac{1}{2}(n+1)(2n+2) = (n+1)^2$, which is the $(n+1)$th square number (A1).
• Award the second mark only if the conclusion explicitly identifies $(n+1)^2$ as a square number / term of P.