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