<p>Find the coefficient of \(x^3\) in the expansion of \((1+2x)^{-3}\) when \(|x| < \frac{1}{2}\).</p>
Step-by-Step Solution
Key Concept: Use the generalized binomial theorem for negative exponents: (1+y)^n = Σ C(n,r)y^r where C(n,r) = n(n-1)(n-2)...(n-r+1)/r!. For (1+2x)^{-3}, identify the coefficient of x^3 by finding the term where the power of x equals 3.
<p><strong>Step 1:</strong> Apply the generalized binomial theorem for (1+y)^n with n = -3 and y = 2x:</p><p>(1+2x)^{-3} = Σ C(-3,r)(2x)^r</p><p><strong>Step 2:</strong> Calculate C(-3,r) for r = 3:</p><p>C(-3,3) = (-3)(-3-1)(-3-2)/(3!) = (-3)(-4)(-5)/6 = -60/6 = -10</p><p><strong>Step 3:</strong> The term containing x^3 is:</p><p>C(-3,3)(2x)^3 = (-10)(8x^3) = -80x^3</p><p><strong>Step 4:</strong> Extract the coefficient of x^3:</p><p>∴ Answer: <strong>-80</strong></p>
Correct Answer: -80