(a) If (17x = 375^{2} – 356^{2}), find the exact value of x.
(b) If (4^{x} = 2^{frac{1}{2}} times 8), find x.
(c) The sum of the first 9 terms of an A.P is 72 and the sum of the next 4 terms is 71, find the A.P.
Explanation
(a) (17x = 375^{2} – 356^{2})
(17x = (375 + 356)(375 – 356))
(17x = (731)(19))
(x = frac{731 times 19}{17} = 817)
(b) (4^{x} = 2^{frac{1}{2}} times 8)
(2^{2x} = 2^{frac{1}{2}} times 2^{3})
(2^{2x} = 2^{3frac{1}{2}})
(implies 2x = 3frac{1}{2} implies x = frac{7}{4})
(c) (S_{n} = frac{n}{2} (2a + (n – 1)d)) (sum of terms of an A.P)
(S_{9} = frac{9}{2} [2a + (9 – 1) d] = frac{9}{2} [2a + 8d])
(72 = 9(a + 4d) implies 8 = a + 4d … (1))
(S_{9 + 4} = S_{13} = frac{13}{2} [2a + (13 – 1)d] = frac{13}{2} [2a + 12d])
(72 + 71 = 143 = 13(a + 6d) implies 11 = a + 6d … (2))
((2) – (1) : 2d = 3 implies d = frac{3}{2})
(a + 4(1frac{1}{2}) = 8 implies a + 6 = 8)
(implies a = 8 – 6 = 2)
(therefore text{The A.P is } 2, 3frac{1}{2}, 5, 6frac{1}{2}, 8,…)