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: Identify the components of the binomial expansion. The binomial theorem states that the (r+1)-th term of (a+b)^n is given by the formula nr a^n-r b^r. In this problem, we have (x+2)^10: • a = x • b = 2 • n = 10 We need to find the 6th term, so r+1 = 6, which means r = 5. Step 2: Substitute the values into the formula. The 6th term is: T_6 = 105 x^10-5 (2)^5 Step 3: Calculate the binomial coefficient and powers. First, calculate 105: 105 = (10!)/(5!(10-5)!) = (10!)/(5!5!) = (10 × 9 × 8 × 7 × 6)/(5 × 4 × 3 × 2 × 1) = 2 × 9 × 2 × 7 = 252 Next, calculate the powers: x^10-5 = x^5 (2)^5 = 32 Step 4: Combine the results to find the 6th term. T_6 = 252 · x^5 · 32 T_6 = 8064x^5 The 6th term of the binomial expansion (x+2)^10 is 8064x^5. That's 2 down. 3 left today — send the next one.