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
It looks like you're asking to find a general formula for U_n for the sequence defined in the original image: U_n+1 = 3 + (1)/(4) U_n with U_1 = 5. This is often the next step after calculating the first few terms. Here's how to find the 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)/(4) U_n. The characteristic equation is r = (1)/(4). The homogeneous solution is U_n^(h) = A ((1)/(4))^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 = 3 + (1)/(4) C Subtract (1)/(4) C from both sides: C - (1)/(4) C = 3 (3)/(4) C = 3 Multiply by (4)/(3): C = 3 × (4)/(3) C = 4 So, the particular solution is U_n^(p) = 4. 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)/(4))^n + 4 Step 4: Determine the constant A using the initial term U_1 = 5. Substitute n=1 and U_1 = 5 into the general solution: 5 = A ((1)/(4))^1 + 4 5 = (A)/(4) + 4 Subtract 4 from both sides: 1 = (A)/(4) Multiply by 4: A = 4 Step 5: Substitute A back into the general solution. U_n = 4 ((1)/(4))^n + 4 This can be simplified: U_n = (4)/(4^n) + 4 U_n = (1)/(4^n-1) + 4 The formula for U_n is U_n = (1)/(4^n-1) + 4. 3 done, 2 left today. You're making progress.