Question

Va rog ajutați-mă cu aceste sume Gauss:

1+3+5+.....+99=

3+7+11+15+......+43=

3+6+9+12+.......+2019=

4+8+12+.......+2020=

5+10+15+20+........+2020=

7+14+21+........+2023=

VA ROG!!! DAU 20 DE PUNCTE!!!!!

Answer 1

$\star)1 + 3 + 5 + ... + 99$

$1 + 3 + 5 + ... + 2n - 1 = {n}^{2}$

$2n - 1 = 99$

$2n = 99 + 1$

$2n = 100 \: | \div 2$

$n = 50$

${n}^{2} = {50}^{2} = 2500$

$\star)3 + 7 + 11 + 15 + ... + 43$

$r = 4 \: (ratia)$

$n = \frac{43 - 3}{4} + 1 = \frac{40}{4} + 1 = 10 + 1 = 11$

$n - nr. \: de \: termeni$

$S = \frac{11(43 + 3)}{2} = \frac{11 \times 46}{2} = 11 \times 23= 253$

$\star)3 + 6 + 9 + 12 + ... + 2019$

$= 3(1 + 2 + 3 + 4 + ... + 673)$

$= 3 \times \frac{673(673 + 1)}{2}$

$= 3 \times \frac{673 \times 674}{2}$

$= 3 \times 673 \times 337$

$= 680403$

$\star)4 + 8 + 12 + ... + 2020$

$= 4(1 + 2 + 3 + ... + 505)$

$= 4 \times \frac{505(505 + 1)}{2}$

$= 4 \times \frac{505 \times 506}{2}$

$= 4 \times 505 \times 253$

$= 511060$

$\star)5 + 10 + 15 + 20 + ... + 2020$

$= 5(1 + 2 + 3 + 4 + ... + 404)$

$= 5 \times \frac{404(404 + 1)}{2}$

$= 5 \times \frac{404 \times 405}{2}$

$= 5 \times 202 \times 405$

$= 409050$

$\star)7 + 14 + 21 + ... + 2023$

$= 7(1 + 2 + 3 + ... + 289)$

$= 7 \times \frac{289(289 + 1)}{2}$

$= 7 \times \frac{289 \times 290}{2}$

$= 7 \times 289 \times 145$

$= 293335$