As given by the question
There are given that the function:
[tex]\sum ^6_{i\mathop=1}7(-\frac{1}{3})^i[/tex]
Now,
To evaluate the given summation:
[tex]\sum ^6_{i\mathop{=}1}7(-\frac{1}{3})^i=7(-\frac{1}{3})^1+7(-\frac{1}{3})^2+7(-\frac{1}{3})^3+7(-\frac{1}{3})^4+7(-\frac{1}{3})^5+7(-\frac{1}{3})^6[/tex]
Then,
[tex]\begin{gathered} \sum ^6_{i\mathop{=}1}7(-\frac{1}{3})^i=7(-\frac{1}{3})^1+7(-\frac{1}{3})^2+7(-\frac{1}{3})^3+7(-\frac{1}{3})^4+7(-\frac{1}{3})^5+7(-\frac{1}{3})^6 \\ =-\frac{7}{3}+\frac{7}{9}-\frac{7}{27}+\frac{7}{81}-\frac{7}{243}+\frac{7}{729} \\ =-2.33+0.77-0.259+0.086-0.0288+0.0096 \end{gathered}[/tex]
Then,
[tex]\begin{gathered} -2.33+0.77-0.259+0.086-0.0288+0.0096=-1.7522 \\ =-1.75 \end{gathered}[/tex]
Hence, the correct option is C.
