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
Alright — let's do this. Step 1: Find the coefficient of x^3 in the expansion of (1+x)^5. The general term in the binomial expansion of (a+b)^n is given by T_k+1 = nk a^n-k b^k. For (1+x)^5, we have a=1, b=x, and n=5. We want the term with x^3, so we set k=3. T_3+1 = 53 (1)^5-3 (x)^3 T_4 = 53 (1)^2 (x)^3 Calculate 53: 53 = (5!)/(3!(5-3)!) = (5!)/(3!2!) = (5 × 4)/(2 × 1) = 10 So the term is 10 · 1 · x^3 = 10x^3. The coefficient of x^3 is 10. The coefficient of x^3 is 10. Step 2: Find the coefficient of x^2 in the expansion of (2x-3)^4. The general term in the binomial expansion of (a+b)^n is given by T_k+1 = nk a^n-k b^k. For (2x-3)^4, we have a=2x, b=-3, and n=4. T_k+1 = 4k (2x)^4-k (-3)^k We want the term with x^2, so the power of x must be 2. From (2x)^4-k, the power of x is 4-k. Set 4-k=2: 4-k=2 k=2 Now substitute k=2 into the general term: T_2+1 = 42 (2x)^4-2 (-3)^2 T_3 = 42 (2x)^2 (-3)^2 Calculate 42: 42 = (4!)/(2!(4-2)!) = (4!)/(2!2!) = (4 × 3)/(2 × 1) = 6 Substitute the values: T_3 = 6 · (4x^2) · (9) T_3 = 6 · 4 · 9 · x^2 T_3 = 216x^2 The coefficient of x^2 is 216. The coefficient of x^2 is 216. Step 3: Find the term independent of x in the expansion of (1-2x). The expression given is (1-2x). The term independent of x is the constant term in this expression. The constant term is 1. The term independent of x is 1. Step 4: Find the sum of coefficients in the expansion of (a-b)^n. To find the sum of coefficients in any polynomial expansion, substitute 1 for each variable. In the expansion of (a-b)^n, we substitute a=1 and b=1. Sum of coefficients = (1-1)^n = 0^n. If n > 0, then 0^n = 0. If n=0, then (a-b)^0 = 1, and the sum of coefficients is 1. Assuming n is a positive integer, which is typical for binomial expansion questions of this type. The sum of coefficients is 0. Step 5: Find the middle term in the expansion of (1-2x)^-4. The expression (1-2x)^-4 represents an infinite series expansion, as the exponent is a negative integer. For example, using the generalized binomial theorem (1+y)^n = 1 + ny + (n(n-1))/(2!)y^2 + Here, y = -2x and n = -4. (1-2x)^-4 = 1 + (-4)(-2x) + ((-4)(-4-1))/(2!)(-2x)^2 + = 1 + 8x + ((-4)(-5))/(2)(4x^2) + = 1 + 8x + 10(4x^2) + = 1 + 8x + 40x^2 + An infinite series does not have a "middle term" because there is no finite number of terms to determine a central position. The expansion of (1-2x)^-4 is an infinite series and therefore has no middle term.