This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the solution to the problem: Step 1: Find the common difference and the first term of the A.P. (Part a) Let the first term be a and the common difference be d. The n-th term of an A.P. is given by T_n = a + (n-1)d. Given: The 5th term is 60: a + (5-1)d = a + 4d = 60 (1) The 10th term is 45: a + (10-1)d = a + 9d = 45 (2) Subtract equation (1) from equation (2): (a + 9d) - (a + 4d) = 45 - 60 5d = -15 d = (-15)/(5) d = -3 Substitute d = -3 into equation (1): a + 4(-3) = 60 a - 12 = 60 a = 60 + 12 a = 72 The common difference is -3 and the first term is 72. Step 2: Determine the least number of terms for the sum to be negative and find the sum (Part b) The sum of n terms of an A.P. is given by S_n = (n)/(2)(2a + (n-1)d). We have a = 72 and d = -3. We want to find the least n such that S_n < 0. S_n = (n)/(2)(2(72) + (n-1)(-3)) S_n = (n)/(2)(144 - 3n + 3) S_n = (n)/(2)(147 - 3n) For S_n < 0: (n)/(2)(147 - 3n) < 0 Since n is the number of terms, n must be a positive integer (n > 0). Therefore, (n)/(2) is positive. For the product to be negative, the other factor must be negative: 147 - 3n < 0 147 < 3n (147)/(3) < n 49 < n The least integer value for n that satisfies n > 49 is n = 50. Now, calculate the sum for n = 50: S_50 = (50)/(2)(147 - 3(50)) S_50 = 25(147 - 150) S_50 = 25(-3) S_50 = -75 The least number of terms is 50 and the sum is -75. Step 3: Determine the 6th term of the G.P. (Part c) The 5th term of the A.P. is T_5 = 60. The 10th term of the A.P. is T_10 = 45. These two terms form two consecutive terms of a G.P. Let the first term of the G.P. be A and the common ratio be R. Assuming the 5th term of the A.P. is the first term of the G.P. and the 10th term of the A.P. is the second term of the G.P.: A = 60 AR = 45 To find the common ratio R: 60R = 45 R = (45)/(60) R = (3)/(4) The n-th term of a G.P. is given by T_n = AR^n-1. We need to find the 6th term of this G.P.: T_6 = AR^6-1 T_6 = AR^5 T_6 = 60 ((3)/(4))^5 T_6 = 60 × (3^5)/(4^5) T_6 = 60 × (243)/(1024) T_6 = (14580)/(1024) T_6 = (3645)/(256) T_6 ≈ 14.23828125 The 6th term of the G.P. is approximately 14.238. Send me the next one 📸