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
Here are the solutions to the problems. 1. The coefficient ratio of the coefficient of x^4 to that of x^3 in the binomial expansion of (1+2x)^n is 3:1. Find the value of n. Step 1: Write the general term of the binomial expansion. The general term T_k+1 for (1+2x)^n is given by: T_k+1 = nk (1)^n-k (2x)^k = nk 2^k x^k Step 2: Find the coefficient of x^3. For x^3, we set k=3. The coefficient of x^3 is n3 2^3 = 8n3. Step 3: Find the coefficient of x^4. For x^4, we set k=4. The coefficient of x^4 is n4 2^4 = 16n4. Step 4: Set up the ratio and solve for n. The ratio of the coefficient of x^4 to that of x^3 is 3:1. (16n)/(4)8n3 = (3)/(1) 2 (n)/(4)n3 = 3 Using the identity (n)/(k)nk-1 = (n-k+1)/(k), with k=4: 2 ((n-4+1)/(4)) = 3 2 ((n-3)/(4)) = 3 (n-3)/(2) = 3 n-3 = 6 n = 9 The value of n is 9. 2. The first three terms of the expansion of (1+ax)^n is 1+2x+(15)/(8)x^2+.... Find the value of a and n. Step 1: Write the first three terms of the binomial expansion of (1+ax)^n. (1+ax)^n = n0 (1)^n (ax)^0 + n1 (1)^n-1 (ax)^1 + n2 (1)^n-2 (ax)^2 + (1+ax)^n = 1 + n