(a) The fourth term of an A.P is 37 and 6th term is 12 more than the fourth term . Find the first and seventh terms.
(b) If \(P = {1, 2, 3, 4}\) and \(Q = {3, 5, 6}\), find (i) \(P \cap Q\) ; (ii) \(P \cup Q\) ; (iii) \((P \cap Q) \cup Q\) ; (iv) \((P \cap Q) \cup P\).
Explanation
(a) \(T_{n} = a + (n - 1)d\) (nth term of an A.P)
\(T_{4} = a + 3d = 37 ... (1)\)
\(T_{6} = a + 5d = 12 + 37 = 49 ... (2)\)
(2) - (1) : \(2d = 12 \implies d = 6\)
\(a + 3d = 37 \implies a + 3(6) = 37\)
\(a + 18 = 37 \implies a = 37 - 18 = 19\)
\(T_{7} = a + 6d = 19 + 6(6) = 19 + 36 = 55\)
(b) \(P = {1, 2, 3, 4} ; Q = {3, 5, 6}\)
(i) \(P \cap Q = {3}\)
(ii) \(P \cup Q = {1, 2, 3, 4, 5, 6}\)
(iii) \((P \cap Q) \cup Q = {3, 5, 6}\)
(iv) \((P \cap Q) \cup P = {1, 2, 3, 4}\)