Answer
$\sum ^{99}_{k=0}\left( -x\right) ^{k}\times \left( k+1\right) $
Work Step by Step
$1-2x+3x^{2}-4x^{3}+5x^{4}+\ldots -100x^{99}=\left( 0+1\right) \times (-x)^{0}+ \left( 1+1\right) \times (-x)^{1}+\left( 2+1\right) \times (-x)^{2}+\left( -1\right) ^{3} \times \left( 3+1\right) \times x^{3}+\left( 4+1\right) \times (-x)^{4}+\ldots \left( 99+1\right) \times (-x)^{99}=\sum ^{99}_{k=0}\left( -x\right) ^{k}\times \left( k+1\right) $
