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
Step 1: Define terms for the linear sequence. Let the first term be a and the common difference be d. The n-th term of a linear sequence is given by a_n = a + (n-1)d. The 4th term is a_4 = a + 3d. The 7th term is a_7 = a + 6d. Step 2: Use the given ratio to form an equation. The 4th term and the 7th term are in the ratio 11:20. (a_4)/(a_7) = (a + 3d)/(a + 6d) = (11)/(20) 20(a + 3d) = 11(a + 6d) 20a + 60d = 11a + 66d 9a = 6d 3a = 2d d = (3)/(2)a Step 3: Use the sum of the 4th and 7th terms to form another equation. The sum of the 4th and 7th terms is 124. a_4 + a_7 = (a + 3d) + (a + 6d) = 124 2a + 9d = 124 Step 4: Solve the system of equations for a and d. Substitute d = (3)/(2)a into the equation 2a + 9d = 124: 2a + 9((3)/(2)a) = 124 2a + (27)/(2)a = 124 (4a + 27a)/(2) = 124 (31a)/(2) = 124 31a = 248 a = (248)/(31) = 8 Now find d: d = (3)/(2)a = (3)/(2)(8) = 12 So, the first term is a=8 and the common difference is d=12. Step 5: Calculate the sum of the first fifteen terms. The sum of the first n terms of an arithmetic sequence is S_n = (n)/(2)(2a + (n-1)d). For n=15: S_15 = (15)/(2)(2(8) + (15-1)12) S_15 = (15)/(2)(16 + 14 × 12) S_15 = (15)/(2)(16 + 168) S_15 = (15)/(2)(184) S_15 = 15 × 92 S_15 = 1380 The sum of the first fifteen terms is 1380. b) Step 1: Set the function equal to y. Given the function f:x 5x + 13, we write it as y = 5x + 13. Step 2: Swap x and y. To find the inverse function, we interchange x and y: x = 5y + 13 Step 3: Solve for y. Subtract 13 from both sides: x - 13 = 5y Divide by 5: y = (x - 13)/(5) The inverse rule of the function is f^-1(x) = (x - 13)/(5).