Q1 A geometric sequence ... has = 27 and a sum to infinity of . u1, u2, u3, u1 (a) Find the common ratio of the geometric sequence. (2) An arithmetic sequence ... is such that = and = v1, v2, v3, v2 u2 v4 u4. (b) Find the greatest value of such that . N (5) (Total 7 marks) (a) = 27 u1 M1 = A1 r (b) = 9 v2 = 1 v4 2d = 8 = (A1) d – –4 = 13 (A1) v1 (2 × 13 4(N 1)) > 0 (accept equality) M1 – – (30 4N) > 0 – 2N) > 0 N(15 – < 7.5 (M1) N = 7 A1 N Note: 13 + 9 + 5 + 1 3 7 11 > 0 = 7 or equivalent N – – – receives full marks. [7] Question 2 (On the assignment) An arithmetic sequence has first term and common difference 0. a d, d ≠ The 3rd, 4th and 7th terms of the arithmetic sequence are the first three terms of a geometric sequence. (a) Show that = . a (3) (b) Show that the 4th term of the geometric sequence is the 16th term of the arithmetic sequence. (5) (Total 8 marks) Question 3
Q1 A geometric sequence u1, u2, u3, ... has u1 = 27 and a sum to infinity of . (a) Find the common ratio of the geometric sequence. (2) An arithmetic sequence v1, v2, v3, ... is such that v2 = u2 and v4 = u4. (b) Find the greatest value of N such that . (5) (Total 7 marks) (a) u1 = 27 M1 r = A1 (b) v2 = 9 v4 = 1 2d = – 8 d = –4 (A1) v1 = 13 (A1) (2 × 13 – 4(N – 1)) > 0 (accept equality) M1 (30 – 4N) > 0 N(15 – 2N) > 0 N < 7.5 (M1) N = 7 A1 Note: 13 + 9 + 5 + 1 – 3 – 7 – 11 > 0 N = 7 or equivalent receives full marks. [7] Question 2 (On the assignment) An arithmetic sequence has first term a and common difference d, d ≠ 0. The 3rd, 4th and 7th terms of the arithmetic sequence are the first three terms of a geometric sequence. (a) Show that a = . (3) (b) Show that the 4th term of the geometric sequence is the 16th term of the arithmetic sequence. (5) (Total 8 marks) Question 3
Past-exam-questions-Solutions.docx