Answer
$\dfrac{23}{12}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given summation expression, $
\displaystyle\sum_{i=2}^5 \dfrac{x_i+1}{x_i+2}
,$ substitute $
i
$ with the values from $
2
$ to $
5
.$ Then substitute the given values for each $x_i$'s
$\bf{\text{Solution Details:}}$
Substituting $
i
$ with the numbers from $
2
$ to $
5
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{x_2+1}{x_2+2}+\dfrac{x_3+1}{x_3+2}+\dfrac{x_4+1}{x_4+2}+\dfrac{x_5+1}{x_5+2}
.\end{array}
Using the given values $x_2=-1, x_3=0, x_4=1,$ and $x_5=2,$ the expression above evaluates to
\begin{array}{l}\require{cancel}
\dfrac{-1+1}{-1+2}+\dfrac{0+1}{0+2}+\dfrac{1+1}{1+2}+\dfrac{2+1}{2+2}
\\\\=
\dfrac{0}{1}+\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}
\\\\=
0+\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}
\\\\=
\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}
\\\\=
\dfrac{6}{12}+\dfrac{8}{12}+\dfrac{9}{12}
\\\\=
\dfrac{23}{12}
.\end{array}
