<p>Find the 7th term of \(\left(\dfrac{4x}{5} - \dfrac{5}{2x}\right)^9\) when expanded in ascending powers of \(x\).</p>
Step-by-Step Solution
Key Concept: The 7th term in ascending powers of x corresponds to the term where the power of x is smallest, not the 7th term in the standard binomial expansion. Use the general term formula and identify which value of r gives the 7th smallest power of x.
<p><strong>Step 1:</strong> Write the general term of (4x/5 - 5/2x)⁹</p><p>Tᵣ₊₁ = ⁹Cᵣ(4x/5)⁹⁻ʳ(-5/2x)ʳ = ⁹Cᵣ · (4/5)⁹⁻ʳ · (-5/2)ʳ · x⁹⁻ʳ · x⁻ʳ</p><p>Tᵣ₊₁ = ⁹Cᵣ · (4/5)⁹⁻ʳ · (-5/2)ʳ · x⁹⁻²ʳ</p><p><strong>Step 2:</strong> Find the power of x in each term: Power = 9 - 2r</p><p>For ascending powers: r = 0, 1, 2, 3, 4, 5, ... gives powers 9, 7, 5, 3, 1, -1, ...</p><p>The 7th term in ascending powers corresponds to the 6th smallest power. When r = 5: power = 9 - 10 = -1</p><p>Actually, for 7th term: we need the 7th value. Counting: r=0(pow 9), r=1(pow 7), r=2(pow 5), r=3(pow 3), r=4(pow 1), r=5(pow -1), r=6(pow -3)</p><p>So the 7th term has r = 6</p><p><strong>Step 3:</strong> Calculate T₇ with r = 6:</p><p>T₇ = ⁹C₆(4/5)³(-5/2)⁶ · x⁻³</p><p>= 84 · (64/125) · (15625/64) · x⁻³</p><p>= 84 · 15625/125 · x⁻³</p><p>= 84 · 125 · x⁻³</p><p>= 10500x⁻³ or 10500/x³</p><p>∴ Answer: <strong>10500x⁻³</strong> (or 10500/x³)</p>
Correct Answer: 10500