The sequence S=i+2i2+3i3+4i4+⋯ up to 100 terms simplifies to, where i=−1

S=i+2i2+3i3+⋯+100i100S⋅i=i2+2i3+⋯+99i100+100i101S(1−i)=i+i2+i3+⋯+i100−100i101S(1−i)=i+i2+i3+i4+i5+i6+i7+i8⇒ S(1−i)=0+0+⋯+i97+i98+i99+i100−100i⇒ S=−100i1−i=−100i(1+i)(1−i)(1+i)⇒ S=−50(i−1)∴ S=50(1−i)