(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})