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Sums of r(1+x)^r and their coefficients

This type of question, asked 2 times: 6 Apr 2024, 22 Jan 2026 S1. All 2 questions below, oldest first.

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  1. Question 1

    6 Apr 2024

    173
    If S(x)=(1+x)+2(1+x)2+3(1+x)3+…S(x) = (1 +x) + 2(1 +x)^{2}+ 3(1 +x)^{3}+ \ldots. +60(1+x)60,x≠0+ 60(1 +x)^{60},x\neq 0, and (60)2S(60)=a(b)b+b(60)^{2}S(60) =a(b)^{b}+b, where a,b∈Na,b \in N, then (a+b)(a + b) equal to

  2. Question 2

    22 Jan 2026 S1

    66
    The coefficient of x48x^{48} in (1+x)+2(1+x)2+3(1+x)3+….+100(1+x)100(1 + x) + 2(1 + x)^{2}+ 3(1 + x)^{3}+ \ldots. + 100(1 + x)^{100} is equal to :

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