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: Expand the first three terms of (1+(x)/(2))^10 using the binomial theorem. The binomial theorem states that (a+b)^n = n0a^n b^0 + n1a^n-1b^1 + n2a^n-2b^2 + . Here, a=1, b=(x)/(2), and n=10. The first term is: 100(1)^10((x)/(2))^0 = 1 · 1 · 1 = 1 The second term is: 101(1)^9((x)/(2))^1 = 10 · 1 · (x)/(2) = 5x The third term is: 102(1)^8((x)/(2))^2 = (10 × 9)/(2 × 1) · 1 · (x^2)/(4) = 45 · (x^2)/(4) = (45)/(4)x^2 So, the expansion of (1+(x)/(2))^10 up to the third term is 1 + 5x + (45)/(4)x^2. Step 2: Multiply the expansion by (1-2x). We need to find the first three terms of (1-2x)(1 + 5x + (45)/(4)x^2 + ). Multiply each term in (1-2x) by the terms in the expansion, keeping only terms up to x^2. Multiply 1 by the expansion: 1 · (1 + 5x + (45)/(4)x^2) = 1 + 5x + (45)/(4)x^2 Multiply -2x by the expansion: -2x · (1 + 5x + (45)/(4)x^2) = -2x - 10x^2 - (45)/(2)x^3 We only need terms up to x^2, so we ignore -(45)/(2)x^3. Step 3: Combine the terms and simplify. Add the results from Step 2: (1 + 5x + (45)/(4)x^2) + (-2x - 10x^2) Combine like terms: 1 + (5x - 2x) + ((45)/(4)x^2 - 10x^2) 1 + 3x + ((45)/(4) - (40)/(4))x^2 1 + 3x + (5)/(4)x^2 The first three terms in the binomial expansion are 1, 3x, and (5)/(4)x^2. The first three terms are 1 + 3x + (5)/(4)x^2. That's 2 down. 3 left today — send the next one.