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: Find the coefficient of x^4 in the expansion of ((1)/(p) + px)^7. The general term in the binomial expansion of (a+b)^n is given by T_r+1 = nr a^n-r b^r. For ((1)/(p) + px)^7, we have a = (1)/(p), b = px, and n = 7. We want the term containing x^4, so we set r=4. T_4+1 = 74 ((1)/(p))^7-4 (px)^4 T_5 = 74 ((1)/(p))^3 p^4 x^4 Calculate the binomial coefficient: 74 = (7!)/(4!(7-4)!) = (7!)/(4!3!) = (7 × 6 × 5)/(3 × 2 × 1) = 35 Substitute this value back into the term: T_5 = 35 · (1)/(p^3) · p^4 x^4 = 35 p^4-3 x^4 = 35p x^4 The coefficient of x^4 in the first expansion is C_1 = 35p. Step 2: Find the coefficient of x^4 in the expansion of ((1)/(2) - 2x)^6. For ((1)/(2) - 2x)^6, we have a = (1)/(2), b = -2x, and n = 6. We want the term containing x^4, so we set r=4. T_4+1 = 64 ((1)/(2))^6-4 (-2x)^4 T_5 = 64 ((1)/(2))^2 (-2)^4 x^4 Calculate the binomial coefficient: 64 = (6!)/(4!(6-4)!) = (6!)/(4!2!) = (6 × 5)/(2 × 1) = 15 Substitute this value back into the term: T_5 = 15 · (1)/(4) · 16 x^4 = 15 · 4 x^4 = 60 x^4 The coefficient of x^4 in the second expansion is C_2 = 60. Step 3: Use the given relationship to find the value of p. The problem states that the coefficient of x^4 in the first expansion is 192 more than the coefficient of x^4 in the second expansion. C_1 = C_2 + 192 Substitute the values of C_1 and C_2: 35p = 60 + 192 35p = 252 Divide by 35 to solve for p: p = (252)/(35) Both 252 and 35 are divisible by 7: p = (252 ÷ 7)/(35 ÷ 7) = (36)/(5) The value of p is (36)/(5) or 7.2. This satisfies the condition p > 0. The final answer is p=(36)/(5). Last free one today — make it count tomorrow, or type /upgrade for unlimited.