(a) A = {1, 2, 5, 7} and B = {1, 3, 6, 7} are subsets of the universal set U = {1, 2, 3,...., 10}. Find (i) \(A'\) ; (ii) \((A \cap B)'\) ; (iii) \((A \cup B)'\) ; (iv) the subsets of B each of which has three elements. (b) Write down the 15th term of the sequence, \(\frac{2}{1 \times 3}, \frac{2}{2 \times 4}, \frac{4}{3 \times 5}, \frac{5}{4 \times 6},...\). (c) An Arithmetic Progression (A.P) has 3 as its first term and 4 as the common difference, (i) write an expression in its simplest form for the nth term ; (ii) find the least term of the A.P that is greater than 100.
Explanation
(a)(i) U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {1, 2, 5, 7} A' = U - A = {3, 4, 6, 8, 9, 10} (ii) \(A \cap B = {1, 2, 5, 7} \cap {1, 3, 6, 7} = {1, 7}\) \((A \cap B)' = {2, 3, 4, 5, 6, 8, 9, 10}\) (iii) \(A \cup B = {1, 2, 5, 7} \cup {1, 3, 6, 7}\) = \({1, 2, 3, 5, 6, 7}\) \((A \cup B)' = {4, 8, 9,10}\) (iv) {1, 3, 6}, {1, 3, 7}, {1, 6, 7} and {3, 6, 7}. (b) The nth term of the sequence is given as \(U_{n} = \frac{n + 1}{n(n + 2)}, n = 1, 2,3\) For the 15th term, \(U_{15} = \frac{16}{15 \times 17}\) (c)(i) \(T_{n} = a + (n - 1)d\) \(T_{n} = 3 + (n - 1)4 = 3 + 4n - 4\) = \(4n - 1\) (ii) \(4n - 1 > 100\) \(4n > 100 + 1 \implies 4n > 101\) \(n > 25.25\) The least term greater than 100 = 26th term.
