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 parameters of the binomial expansion. The given expansion is ((a)/(2) + 2)^8. Here, n=8. The number of terms in the expansion is n+1 = 8+1 = 9. Step 2: Determine the position of the middle term. Since there are 9 terms, the middle term is the (9+1)/(2) = 5^th term. For the general term T_r+1, we have r+1 = 5, so r=4. Step 3: Write the formula for the general term and substitute the values. The general term in the expansion of (x+y)^n is given by T_r+1 = nr x^n-r y^r. In this case, x = (a)/(2), y = 2, n=8, and r=4. T_5 = 84 ((a)/(2))^8-4 (2)^4 T_5 = 84 ((a)/(2))^4 (2)^4 T_5 = 84 (a^4)/(2^4) · 2^4 T_5 = 84 a^4 Step 4: Calculate the binomial coefficient 84. 84 = (8!)/(4!(8-4)!) = (8!)/(4!4!) = (8 × 7 × 6 × 5)/(4 × 3 × 2 × 1) 84 = (1680)/(24) 84 = 70 Step 5: Set up the equation using the given value of the middle term. We are given that the middle term is 1120. 70 a^4 = 1120 Step 6: Solve for a. a^4 = (1120)/(70) a^4 = 16 Take the fourth root of both sides: a = ± [4]16 a = ± 2 The value of a is ± 2. Send me the next one 📸