Question

Determinati conjugatul numarului complex care reprezinta rezultatul calculului 1+2i+3i²+4i³+...+10i^9.

Answer 1

[tex]Notam\ cu\ z\ toata\ suma:\\ z=1+2i+3i^2+4i^3+...+10i^9 |\cdot i\\ iz=i+2i^2+3i^3+.....+10i^{10}\\ -------------\\ z-iz=1+i+i^2+...+i^9-10i^{10}\\ z(1-i)=(1+i+i^2)+i^3(1+i+i^2)+i^6(1+i+i^2)+i^9-10i^{10}\\ z(1-i)=i+i^4+i^7+i^9-10i^{10}\\ z(1-i)=i+1-i+i+10\\ z(1-i)=i+11\\ z=\frac{i+11}{1-i}=\frac{(i+11)(1-i)}{2}=\frac{i+11+1-11i}{2}=\frac{12-10i}{2}=6-5i\\ Asadar:\boxed{z=6-5i}\\ \overline{z}=6+5i[/tex]