To the basics of summation

This can be expressed as $\sum _{ n=1 }^{ 100 }{ n\left( 101-n \right) }$

This can be simplified to $\sum _{ n=1 }^{ 100 }{ 101n-{ n }^{ 2 } }$

Splitting the summation, $\sum _{ n=1 }^{ 100 }{ 101n } -\sum _{ n=1 }^{ 100 }{ { n }^{ 2 } }$

Taking out 101 from the first summation, $101\sum _{ n=1 }^{ 100 }{ n } -\sum _{ n=1 }^{ 100 }{ { n }^{ 2 } }$

Solving the summations, $101\frac { 100\left( 100+1 \right) }{ 2 } -\frac { 100\left( 100+1 \right) \left( 2\times 100+1 \right) }{ 6 }$

which can be solved to get $\boxed{171700}$