Answer
$\dfrac{728}{9}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given summation expression, $
\displaystyle\sum_{i=-2}^3 2(3)^i
,$ substitute $
i
$ with the values from $
-2
$ to $
3
$ and then simplify the expression.
$\bf{\text{Solution Details:}}$
Substituting $
i
$ with the numbers from $
-2
$ to $
3
,$ the given expression evaluates to
\begin{array}{l}\require{cancel}
2(3)^{-2}+2(3)^{-1}+2(3)^0+2(3)^1+2(3)^2+2(3)^3
\\\\=
\dfrac{2}{3^2}+\dfrac{2}{3^1}+2(1)+2(3)+2(9)+2(27)
\\\\=
\dfrac{2}{9}+\dfrac{2}{3}+2+6+18+54
\\\\=
\dfrac{2}{9}+\dfrac{6}{9}+\dfrac{18}{9}+\dfrac{54}{9}+\dfrac{162}{9}+\dfrac{486}{9}
\\\\=
\dfrac{728}{9}
.\end{array}