Answer
$$28$$
Work Step by Step
$$\eqalign{
& \sum\limits_{i = 1}^7 {{{\left( { - 1} \right)}^{i + 1}} \cdot {i^2}} \cr
& {\text{Write each of the seven terms}},{\text{ and then evaluate the sum}} \cr
& = {\left( { - 1} \right)^{1 + 1}} \cdot {\left( 1 \right)^2} + {\left( { - 1} \right)^{2 + 1}} \cdot {\left( 2 \right)^2} + {\left( { - 1} \right)^{3 + 1}} \cdot {\left( 3 \right)^2} + {\left( { - 1} \right)^{4 + 1}} \cdot {\left( 4 \right)^2} \cr
& \,\,\, + {\left( { - 1} \right)^{5 + 1}} \cdot {\left( 5 \right)^2} + {\left( { - 1} \right)^{6 + 1}} \cdot {\left( 6 \right)^2} + {\left( { - 1} \right)^{7 + 1}} \cdot {\left( 7 \right)^2} \cr
& {\text{Simplifying}} \cr
& = \left( 1 \right) \cdot \left( 1 \right) + \left( { - 1} \right) \cdot \left( 4 \right) + \left( 1 \right) \cdot \left( 9 \right) + \left( { - 1} \right) \cdot \left( {16} \right) + \left( 1 \right) \cdot \left( {25} \right) \cr
& \,\,\,\,\, + \left( { - 1} \right) \cdot \left( {36} \right) + \left( 1 \right) \cdot \left( {49} \right) \cr
& = 1 - 4 + 9 - 16 + 25 - 36 + 49 \cr
& = 28 \cr} $$
