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
You're on a roll — Step 1: Write the general term for the binomial expansion of (a + (x)/(2))^4. The general term T_r+1 is given by: T_r+1 = nr A^n-r B^r In this case, A=a, B=(x)/(2), and n=4. T_r+1 = 4r a^4-r ((x)/(2))^r T_r+1 = 4r a^4-r (x^r)/(2^r) Step 2: Find the term containing x^3. For the term containing x^3, we need r=3. Substitute r=3 into the general term: T_3+1 = 43 a^4-3 (x^3)/(2^3) T_4 = 43 a^1 (x^3)/(8) Step 3: Calculate the binomial coefficient 43. 43 = (4!)/(3!(4-3)!) = (4!)/(3!1!) = (4 × 3 × 2 × 1)/((3 × 2 × 1)(1)) = 4 Step 4: Substitute the value of 43 back into the term. T_4 = 4 · a · (x^3)/(8) T_4 = (4a)/(8) x^3 T_4 = (a)/(2) x^3 Step 5: Identify the coefficient of x^3 and set it equal to 1. The coefficient of x^3 is (a)/(2). Given that the coefficient of x^3 is 1: (a)/(2) = 1 Step 6: Solve for a. a = 1 × 2 a = 2 The value of a is 2. What's next?