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  4. The sequence S = i +2 i 2+3 i 3+ ldots ldots upto 100 terms simplifies to where i =√-1

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Q. The sequence $S = i +2 i ^2+3 i ^3+\ldots \ldots$ upto $100$ terms simplifies to where $i =\sqrt{-1}$

Complex Numbers and Quadratic Equations

Solution:

$S=i+2 i^2+3 i^3+\ldots+100\, i^{100}$
$i \,S=i^2+2 i^3+\ldots+100\, i^{101}$
$S(1-i)=i+i^2+i^3+\ldots+i^{100}-100\, i^{101}$
$S=\frac{-100 i}{1-i}=\frac{-100 i(1+i)}{2}=-50(i-1)=50(1-i)$