[tex] \star)1 + 3 + 5 + ... + 99[/tex]
[tex]1 + 3 + 5 + ... + 2n - 1 = {n}^{2} [/tex]
[tex]2n - 1 = 99[/tex]
[tex]2n = 99 + 1[/tex]
[tex]2n = 100 \: | \div 2[/tex]
[tex]n = 50[/tex]
[tex] {n}^{2} = {50}^{2} = 2500[/tex]
[tex] \star)3 + 7 + 11 + 15 + ... + 43[/tex]
[tex]r = 4 \: (ratia)[/tex]
[tex]n = \frac{43 - 3}{4} + 1 = \frac{40}{4} + 1 = 10 + 1 = 11[/tex]
[tex]n - nr. \: de \: termeni[/tex]
[tex]S = \frac{11(43 + 3)}{2} = \frac{11 \times 46}{2} = 11 \times 23= 253[/tex]
[tex] \star)3 + 6 + 9 + 12 + ... + 2019[/tex]
[tex] = 3(1 + 2 + 3 + 4 + ... + 673)[/tex]
[tex] = 3 \times \frac{673(673 + 1)}{2} [/tex]
[tex] = 3 \times \frac{673 \times 674}{2} [/tex]
[tex] = 3 \times 673 \times 337[/tex]
[tex] = 680403[/tex]
[tex] \star)4 + 8 + 12 + ... + 2020[/tex]
[tex] = 4(1 + 2 + 3 + ... + 505)[/tex]
[tex] = 4 \times \frac{505(505 + 1)}{2} [/tex]
[tex] = 4 \times \frac{505 \times 506}{2} [/tex]
[tex] = 4 \times 505 \times 253[/tex]
[tex] = 511060[/tex]
[tex] \star)5 + 10 + 15 + 20 + ... + 2020[/tex]
[tex] = 5(1 + 2 + 3 + 4 + ... + 404)[/tex]
[tex] = 5 \times \frac{404(404 + 1)}{2} [/tex]
[tex] = 5 \times \frac{404 \times 405}{2} [/tex]
[tex] = 5 \times 202 \times 405[/tex]
[tex] = 409050[/tex]
[tex] \star)7 + 14 + 21 + ... + 2023[/tex]
[tex] = 7(1 + 2 + 3 + ... + 289)[/tex]
[tex] = 7 \times \frac{289(289 + 1)}{2} [/tex]
[tex] = 7 \times \frac{289 \times 290}{2} [/tex]
[tex] = 7 \times 289 \times 145[/tex]
[tex] = 293335[/tex]
