Answer
$28$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given summation expression, $
\displaystyle\sum_{i=1}^7 (-1)^{i+1}\cdot i^2
,$ substitute $
i
$ with the values from $
1
$ to $
7
$ and then simplify the expression.
$\bf{\text{Solution Details:}}$
Substituting $
i
$ with the numbers from $
1
$ to $
7
,$ the given expression evaluates to
\begin{array}{l}\require{cancel}
[(-1)^{1+1}\cdot 1^2]+[(-1)^{2+1}\cdot 2^2]+[(-1)^{3+1}\cdot 3^2]+[(-1)^{4+1}\cdot 4^2]+[(-1)^{5+1}\cdot 5^2]+[(-1)^{6+1}\cdot 6^2]+[(-1)^{7+1}\cdot 7^2]
\\\\=
[(-1)^{2}\cdot 1]+[(-1)^{3}\cdot 4]+[(-1)^{4}\cdot 9]+[(-1)^{5}\cdot 16]+[(-1)^{6}\cdot 25]+[(-1)^{7}\cdot 36]+[(-1)^{8}\cdot 49]
\\\\=
[(1)\cdot 1]+[(-1)\cdot 4]+[(1)\cdot 9]+[(-1)\cdot 16]+[(1)\cdot 25]+[(-1)\cdot 36]+[(1)\cdot 49]
\\\\=
1-4+9-16+25-36+49
\\\\=
28
.\end{array}