60 B1 2 B1 for correct Tn formula M n(n Tn = + 1) M1 T2n + 1 = (2n + 1)(2n + 1 + 1) M1 for substituting 2n + 1 = (2n + 1)(2n + 2) = (2n + 1)(n + 1) = 2n2 + 3n + 1 (1) …….. A1 A1 cao Tn + 1 = (n + 1)(n + 1 + 1) M1 n M1 for substituting + 1 = (n + 1)(n + 2) A1 A1 cao = (n2 + 3n + 2) …….(2) B1 B1 for subtracting each equation So T2n + 1 Tn + 1 – n2 n = 2n2 + 3n + 1 1 B1 – – – B1 for clear full explanation of proving the final connection n2 n = + 7 n) = (n2 + 61 B1 2 B1 for Tn formula M n(n Tn = + 1) n(n = (n + 1) but Tn = + 1) B3 B1 for numerator expansion = = 3Tn as given B1 for denominator expansion B1 for cancelling == n2 n But + 2 factorises to – (n 1)(n + 2) – B1 for correct factorisation B1 So final expression is B1 for fully clear correct proof with no mathematical B1 notational errors 6
60 Tn = n(n + 1) T2n + 1 = (2n + 1)(2n + 1 + 1) = (2n + 1)(2n + 2) = (2n + 1)(n + 1) = 2n2 + 3n + 1 …….. (1) Tn + 1 = (n + 1)(n + 1 + 1) = (n + 1)(n + 2) = (n2 + 3n + 2) …….(2) So T2n + 1 – Tn + 1 = 2n2 + 3n + 1 – n2 – n – 1 = n2 + n = (n2 + n) = (n + 1) but Tn = n(n + 1) = 3Tn as given B1 M1 A1 M1 A1 B1 B1 2 B1 for correct Tn formula M1 for substituting 2n + 1 A1 cao M1 for substituting n + 1 A1 cao B1 for subtracting each equation B1 for clear full explanation of proving the final connection M 7 61 Tn = n(n + 1) = == But n2 + n – 2 factorises to (n – 1)(n + 2) So final expression is B1 B3 B1 B1 2 B1 for Tn formula B1 for numerator expansion B1 for denominator expansion B1 for cancelling B1 for correct factorisation B1 for fully clear correct proof with no mathematical notational errors M 6
Higher_Skills_Algebra_MarkScheme_AQA.docx