(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.
Show Answer Show Explanation Explanation (a) \(17x = 375^{2} - 356^{2}\) \(17x = (375 + 356)(375 - 356)\) \(17x = (731)(19)\) \(x = \frac{731 \times 19}{17} = 817\) (b) \(4^{x} = 2^{\frac{1}{2}} \times 8\) \(2^{2x} = 2^{\frac{1}{2}} \times 2^{3}\) \(2^{2x} = 2^{3\frac{1}{2}}\) \(\implies 2x = 3\frac{1}{2} \implies x = \frac{7}{4}\) (c) \(S_{n} = \frac{n}{2} (2a + (n - 1)d)\) (sum of terms of an A.P) \(S_{9} = \frac{9}{2} [2a + (9 - 1) d] = \frac{9}{2} [2a + 8d]\) \(72 = 9(a + 4d) \implies 8 = a + 4d ... (1)\) \(S_{9 + 4} = S_{13} = \frac{13}{2} [2a + (13 - 1)d] = \frac{13}{2} [2a + 12d]\) \(72 + 71 = 143 = 13(a + 6d) \implies 11 = a + 6d ... (2)\) \((2) - (1) : 2d = 3 \implies d = \frac{3}{2}\) \(a + 4(1\frac{1}{2}) = 8 \implies a + 6 = 8\) \(\implies a = 8 - 6 = 2\) \(\therefore \text{The A.P is } 2, 3\frac{1}{2}, 5, 6\frac{1}{2}, 8,...\)