1. A geometric sequence u1, u2, u3, ... has u1 = 27 and a sum to infinity of 2 (a) Find the common ratio of the geometric sequence. An arithmetic sequence v1, V2, V3, ... is such that v2 = U2 and v4 = u4. (b) Find the greatest value of N such that N Evn > 0. n =1 2. 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. 81 . (2) (5) (Total 7 marks) (a) Show that a = d . (3) (b) Show that the 4th term of the geometric sequence is the 16th term of the arithmetic sequence. (Total 8 marks) (5) 3. Two players, A and B, alternately throw a fair six-sided dice, with A starting, until one of them obtains a six. Find the probability that A obtains the first six. (Total 7 marks) 4. The mean of the first ten terms of an arithmetic sequence is 6. The mean of the first twenty terms of the arithmetic sequence is 16. Find the value of the 15th term of the sequence. (Total 6 marks) IB Questionbank Mathematics Higher Level 3rd edition 1
5. The sum, Sn, of the first n terms of a geometric sequence, whose nt" term is un, is given by , where a > 0. (a) Find an expression for un- S,= " -" (b) Find the first term and common ratio of the sequence. (c) Consider the sum to infinity of the sequence. (i) Determine the values of a such that the sum to infinity exists. (ii) Find the sum to infinity when it exists. (2) (4) (Total 8 marks) (2) 6. (a) Consider the set of numbers a, 2a, 3a, ... , na where a and n are positive integers. (i) Show that the expression for the mean of this set is a(n+1) . 2 (ii) Let a = 4. Find the minimum value of n for which the sum of these numbers exceeds its mean by more than 100. (6) (b) Consider now the set of numbers x1, ... , Xm, y1, ... , y1, ... , yn where xi = 0 for i = 1, ... , m and yi = 1 for i = 1, ... , n. n and the standard deviation S by Vmn (i) Show that the mean M of this set is given by m+n . m + n (ii) Given that M = S, find the value of the median. (11) (Total 17 marks) IB Questionbank Mathematics Higher Level 3rd edition 2
7. (a) The sum of the first six terms of an arithmetic series is 81. The sum of its first eleven terms is 231. Find the first term and the common difference. (6) (b) The sum of the first two terms of a geometric series is 1 and the sum of its first four