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Find the coefficient of x¹⁰¹ in the expression S = (5+x)⁵⁰⁰ + x*(5+x)⁴⁹⁹ + x²*(5+x)⁴⁹⁸ +... + x⁵⁰⁰, where x > 0.

1. ⁵⁰¹C101 * 5³⁹⁹
2. ⁵⁰¹C101 * 5⁴⁰⁰
3. ⁵⁰¹C100 * 5⁴⁰⁰
4. ⁵⁰⁰C101 * 5³⁹⁹

Correct answer: ⁵⁰¹C101 * 5⁴⁰⁰

Solution

The sum S is a GP: S = sumₖ₌₀⁵⁰⁰ x^k*(5+x)^(500-k). Multiplying by (5+x-x)=5: 5S = (5+x)⁵⁰¹ - x⁵⁰¹. So S = [(5+x)⁵⁰¹ - x⁵⁰¹]/5. The coefficient of x¹⁰¹ in (5+x)⁵⁰¹ is C(501,101)*5⁴⁰⁰. Dividing by 5 gives C(501,101)*5³⁹⁹. Hmm, but x⁵⁰¹ has no x¹⁰¹ term, so coefficient of x¹⁰¹ in S = C(501,101)*5⁴⁰⁰/5... Let me redo: 5*S = (5+x)⁵⁰¹ - x⁵⁰¹. Coefficient of x¹⁰¹ in (5+x)⁵⁰¹ = C(501,101)*5^(501-101) = C(501,101)*5⁴⁰⁰. Coefficient of x¹⁰¹ in x⁵⁰¹ = 0. So coeff of x¹⁰¹ in 5S = C(501,101)*5⁴⁰⁰, hence coeff in S = C(501,101)*5⁴⁰⁰/5 = C(501,101)*5³⁹⁹. Answer should be ⁵⁰¹C101 * 5³⁹⁹.

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