<p>Find the 6th term in the expansion of \((2x^2 - 1/3x^2)^{10}\).</p>
Step-by-Step Solution
Key Concept: The 6th term corresponds to r=5 in the general term T_{r+1} = C(n,r)(a)^{n-r}(b)^r. Substitute a=2x², b=-1/(3x²), n=10, and r=5 to get T_6.
<p><strong>Step 1:</strong> Identify parameters: n=10, a=2x², b=-1/(3x²). For the 6th term, r=5 (since T_{r+1} is the general term).</p><p><strong>Step 2:</strong> Apply general term formula: T_6 = C(10,5)(2x²)^{10-5}(-1/3x²)^5</p><p><strong>Step 3:</strong> Calculate C(10,5) = 252</p><p><strong>Step 4:</strong> Expand: T_6 = 252 · (2x²)^5 · (-1/3x²)^5 = 252 · 32x^{10} · (-1/243x^{10})</p><p><strong>Step 5:</strong> Simplify: T_6 = 252 · 32 · (-1/243) = -8064/243 = -896/27</p><p>∴ Answer: <strong>-896/27</strong></p>
Correct Answer: -896/27