(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.