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Find the greatest term in the expansion of sqrt(3) * (1 + 1/sqrt(3))²⁰.

1. 25840/9
2. 24840/9
3. 26840/9
4. None of these

Correct answer: None of these

Solution

The general term is T_(r+1) = sqrt(3)*C(20,r)*(1/sqrt(3))^r = C(20,r)*(1/sqrt(3))^(r-1) = C(20,r)*3^((1-r)/2). For r=7: T₈ = C(20,7)*3^(-3) = 77520/27 = 2871.1... Compute C(20,7) = 77520. T₈ = sqrt(3)*77520*(1/sqrt(3))⁷ = sqrt(3)*77520/3^(7/2) = 77520/3³ = 77520/27 = 2871.11. None of the listed options match exactly; however computing carefully: 25840/9 = 2871.1... = 77520/27. Check: 25840/9 = 2871.1 and 77520/27 = 2871.1. Indeed 25840/9 = 77520/27? 25840*3=77520. Yes! So T₈ = 77520/27 = 25840/9.

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