Explicație pas cu pas:
a)
[tex]( - 1) \cdot {( - 1)}^{2} \cdot {( - 1)}^{3} \cdot ... \cdot {( - 1)}^{2019} = \\[/tex]
[tex]= {( - 1)}^{1 + 2 + 3 + ... + 2019} = {( - 1)}^{ \frac{2019 \cdot 2020}{2}} \\ = {( - 1)}^{2019 \cdot 1010} = {\Big({( - 1)}^{2019 \cdot 505}\Big)}^{2} = \bf 1\\[/tex]
b)
[tex]( - 1) + {( - 1)}^{2} + {( - 1)}^{3} + ... + {( - 1)}^{2019} = \\[/tex]
[tex]= ( - 1) + 1 + ( - 1) + 1 + ... + 1 + ( - 1) \\ [/tex]
[tex]= - 1 + \underbrace{(1 - 1) + (1 - 1) + ... + (1 - 1)}_{1009}\\ [/tex]
[tex]= - 1 + \underbrace{0 + 0 + ... + 0}_{1009} = \bf -1[/tex]
c)
[tex]( - 1) - {( - 1)}^{2} - {( - 1)}^{3} - ... - {( - 1)}^{2018} - {( - 1)}^{2019} = \\ [/tex]
[tex]= - 1 - 1 - ( - 1) - 1 - ... - 1 - ( - 1)\\[/tex]
[tex]= - 1 - 1 + 1 - 1 + ... - 1 + 1[/tex]
[tex]= - 1 + \underbrace{( - 1 + 1) + ( - 1 + 1) + ... + ( - 1 + 1)}_{1009}\\ [/tex]
[tex]= - 1 + \underbrace{0 + 0 + ... + 0}_{1009} = \bf -1[/tex]
d)
[tex]( - 1) + 2 \cdot {( - 1)}^{2} + 3 \cdot {( - 1)}^{3} + 4 \cdot {( - 1)}^{4} + ... + 2017 \cdot {( - 1)}^{2017} + 2018 \cdot {( - 1)}^{2018} + 2019 \cdot {( - 1)}^{2019} = \\[/tex]
[tex]= - 1 + 2 - 3 + 4 - ... - 2017 + 2018 - 2019 \\ [/tex]
[tex]= \underbrace{(- 1 + 2) + ( - 3 + 4) - ... + ( - 2017 + 2018)}_{1009} - 2019 \\ [/tex]
[tex] = \underbrace{1 + 1 + ... + 1}_{1009} - 2019 = 1009 - 2019 = \bf - 1010 \\ [/tex]
