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 how to solve parts (b) and (c) for the sequence defined by U_n+1 = 1 + (1)/(2) U_n. We will express the formulas in terms of the initial term U_1, as it is not specified for this new recurrence relation. b) Find a formula for U_n using the general method. Step 1: Find the homogeneous solution. The homogeneous part of the recurrence relation is U_n+1 = (1)/(2) U_n. The characteristic equation is r = (1)/(2). Thus, the homogeneous solution is U_n^(h) = A ((1)/(2))^n, where A is a constant. Step 2: Find the particular solution. Assume a constant particular solution U_n^(p) = C. Substitute this into the original recurrence relation: C = 1 + (1)/(2) C C - (1)/(2) C = 1 (1)/(2) C = 1 C = 2 So, the particular solution is U_n^(p) = 2. Step 3: Combine to find the general solution. The general solution is the sum of the homogeneous and particular solutions: U_n = U_n^(h) + U_n^(p) U_n = A ((1)/(2))^n + 2 Step 4: Determine the constant A using the initial term U_1. For n=1: U_1 = A ((1)/(2))^1 + 2 U_1 = (A)/(2) + 2 (A)/(2) = U_1 - 2 A = 2(U_1 - 2) Step 5: Substitute A back into the general