(i) The sum of the first three terms of an arithmetie sequence is 36 and their product is 1140 . Determine the three terms. (o) A certain sequence has a constant second difference of 8 . The sixth term of the sequence is 172 and the tenth term is 452 . Determine the general term. (p) The sum of the first \( n \) terms of a series is given by \( S_{n}=\frac{n}{4}(7-2 n) \). (1) Calculate the sum of the first 30 terms of this series. (1)

(i) Let's denote the first term of the arithmetic sequence as \(a\) and the common difference as \(d\). Given that the sum of the first three terms is 36, we can write the equation: \[ a + (a + d) + (a + 2d) = 36 \] \[ 3a + 3d = 36 \] \[ a + d = 12 \] Also, the product of the first three terms is 1140, so we have: \[ a(a + d)(a + 2d) = 1140 \] Now, we need to solve these equations to find the values of \(a\) and \(d\). (ii) For the sequence with a constant second difference of 8, we can denote the first term as \(a\) and the common difference as \(d\). Given that the sixth term is 172 and the tenth term is 452, we can write the equations: \[ a + 5d = 172 \] \[ a + 9d = 452 \] We can solve these equations to find the values of \(a\) and \(d\). (iii) For the series with the sum of the first \(n\) terms given by \(S_n = \frac{n}{4}(7 - 2n)\), we can calculate the sum of the first 30 terms by substituting \(n = 30\) into the formula. Calculate the value by following steps: - step0: Calculate: \(\frac{30}{4}\left(7-2\times 30\right)\) - step1: Multiply the numbers: \(\frac{30}{4}\left(7-60\right)\) - step2: Subtract the numbers: \(\frac{30}{4}\left(-53\right)\) - step3: Reduce the fraction: \(\frac{15}{2}\left(-53\right)\) - step4: Multiply the numbers: \(-\frac{15}{2}\times 53\) - step5: Multiply: \(-\frac{15\times 53}{2}\) - step6: Multiply: \(-\frac{795}{2}\) (i) The first term of the arithmetic sequence is \(a = 12 - d\). Substitute \(a = 12 - d\) into the equation \(a(a + d)(a + 2d) = 1140\): \((12 - d)(12 - d + d)(12 - d + 2d) = 1140\) \(12(12 - d)(12 + d) = 1140\) \(12(144 - d^2) = 1140\) \(1728 - 12d^2 = 1140\) \(12d^2 = 588\) \(d^2 = 49\) \(d = \pm 7\) Substitute \(d = 7\) into \(a + d = 12\): \(a + 7 = 12\) \(a = 5\) Substitute \(d = -7\) into \(a + d = 12\): \(a - 7 = 12\) \(a = 19\) Therefore, the three terms of the arithmetic sequence are 5, 12, and 19. (ii) The first term of the sequence is \(a = 172 - 5d\). Substitute \(a = 172 - 5d\) into the equation \(a + 9d = 452\): \(172 - 5d + 9d = 452\) \(4d = 280\) \(d = 70\) Substitute \(d = 70\) into \(a + 5d = 172\): \(a + 5(70) = 172\) \(a + 350 = 172\) \(a = -178\) Therefore, the general term of the sequence is \(a_n = -178 + 70(n - 1)\). (iii) The sum of the first 30 terms of the series is \(-397.5\).