Question 1209783
<br>
{{{1+i/3-1/9-i/27+1/81}}}<br>
Combine like terms -- i.e., combine the real terms and combine the imaginary terms.<br>
{{{(1-1/9+1/81)+i(1/3-1/27)}}}
{{{(81/81-9/81+1/81)+i(9/27-1/27)}}}
{{{(73/81)+i(8/27)}}}<br>
ANSWER: (73/81)+(8/27)i<br>
It is possible that the sequence was supposed to be an infinite sequence instead of a finite one.  In that case....<br>
Sum = (first term)/(1-common difference)<br>
{{{1/(1-i/3)=1/((3-i)/3)=3/(3-i)=(3(3+i))/10=(9+3i)/10}}}<br>
ANSWER: (9+3i)/10<br>
Alternatively, we can find the infinite sums of the real and imaginary parts separately.<br>
Real parts: first term 1, common ratio (-1/9)<br>
Sum = {{{1/(1-(-1/9))=1/(1+1/9)=1/(10/9)=9/10}}}<br>
Imaginary parts: first term i/3, common ratio (-1/9)<br>
{{{(i/3)/(1-(-1/9))=(i/3)/(1+1/9)=(i/3)/(10/9)=(i/3)(9/10)=3i/10}}}<br>
ANSWER: (9/10)+(3/10)i<br>