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 — To expand (1 - (1)/(2)x)^6 to the fourth term, we use the binomial theorem, which states that: (a+b)^n = n0a^n b^0 + n1a^n-1b^1 + n2a^n-2b^2 + n3a^n-3b^3 + In this case, a=1, b=-(1)/(2)x, and n=6. Step 1: Calculate the first term (k=0). Term 1 = 60(1)^6(-(1)/(2)x)^0 Term 1 = 1 × 1 × 1 = 1 Step 2: Calculate the second term (k=1). Term 2 = 61(1)^5(-(1)/(2)x)^1 Term 2 = 6 × 1 × (-(1)/(2)x) = -3x Step 3: Calculate the third term (k=2). Term 3 = 62(1)^4(-(1)/(2)x)^2 62 = (6 × 5)/(2 × 1) = 15 Term 3 = 15 × 1 × ((1)/(4)x^2) = (15)/(4)x^2 Step 4: Calculate the fourth term (k=3). Term 4 = 63(1)^3(-(1)/(2)x)^3 63 = (6 × 5 × 4)/(3 × 2 × 1) = 20 Term 4 = 20 × 1 × (-(1)/(8)x^3) = -(20)/(8)x^3 = -(5)/(2)x^3 Step 5: Combine the terms. The expansion to the fourth term is the sum of these terms: 1 - 3x + (15)/(4)x^2 - (5)/(2)x^3 The expansion of (1 - (1)/(2)x)^6 to the fourth term is 1 - 3x + (15)/(4)x^2 - (5)/(2)x^3. That's 2 down. 3 left today — send the next one.