<p>Find the fourth term in the expansion of \((1-2x)^{-3/4}\) if \(|x| < \frac{1}{2}\).</p>
Step-by-Step Solution
Key Concept: Use the generalized binomial theorem for negative/fractional exponents: (1+a)^n = 1 + na + n(n-1)a²/2! + n(n-1)(n-2)a³/3! + ... The fourth term corresponds to the coefficient of the third power of the variable in the expansion.
<p><strong>Step 1:</strong> Apply the generalized binomial theorem with n = -3/4 and a = -2x:</p><p>(1-2x)^(-3/4) = Σ C(-3/4, r)(-2x)^r where C(n,r) = n(n-1)(n-2)...(n-r+1)/r!</p><p><strong>Step 2:</strong> Find the fourth term (r = 3):</p><p>Fourth term = C(-3/4, 3)(-2x)³</p><p><strong>Step 3:</strong> Calculate C(-3/4, 3):</p><p>C(-3/4, 3) = (-3/4)(-3/4 - 1)(-3/4 - 2) / 3!</p><p>= (-3/4)(-7/4)(-11/4) / 6</p><p>= -231/64 / 6 = -231/384 = -77/128</p><p><strong>Step 4:</strong> Multiply by (-2x)³ = -8x³:</p><p>Fourth term = (-77/128)(-8x³) = 616x³/128 = <strong>77x³/16</strong></p><p>∴ Answer: <strong>77x³/16</strong></p>
Correct Answer: 77