<p>Which term in the expansion of \((2 - 3x)^{19}\) has algebraically the least coefficient?</p>
<p>(1) \(10^{\text{th}}\)</p>
<p>(2) \(11^{\text{th}}\)</p>
<p>(3) \(12^{\text{th}}\)</p>
<p>(4) \(13^{\text{th}}\)</p>
Step-by-Step Solution
Key Concept: The coefficient of the rth term in (a + b)^n is C(n,r-1)·a^(n-r+1)·b^(r-1). For (2 - 3x)^19, coefficients involve alternating signs and increasing powers of 3, so the algebraically smallest (most negative) coefficient occurs when the negative term's magnitude is largest.
<p><strong>Step 1:</strong> General term in (2 - 3x)^19 is: T(r+1) = C(19,r)·2^(19-r)·(-3x)^r = C(19,r)·2^(19-r)·(-3)^r·x^r</p><p><strong>Step 2:</strong> The coefficient is: a_r = C(19,r)·2^(19-r)·(-3)^r</p><p><strong>Step 3:</strong> For r even, coefficient is positive. For r odd, coefficient is negative (most negative when |coefficient| is largest).</p><p><strong>Step 4:</strong> Magnitude of coefficient: |a_r| = C(19,r)·2^(19-r)·3^r. This increases with r (since 3 > 2).</p><p><strong>Step 5:</strong> The algebraically smallest coefficient occurs at the largest odd value of r, which is r = 19.</p><p><strong>Step 6:</strong> T(20) = C(19,19)·2^0·(-3)^19·x^19 = 1·1·(-3)^19·x^19</p><p><strong>Step 7:</strong> The 20th term has the algebraically least coefficient of -3^19.</p><p>∴ Answer: <strong>20th term (D)</strong></p>
Correct Answer: D