(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.
(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.
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