Question

s=127 + 129 +131 ...+2017 +2019​

Answer 1

Metoda I:

S = 127+129+131+...+2017+2019

S = 63+64+65+...+1008+1009+

   +64+65+66+....+1009+1010

S = 63+64+65+...+1008+1009+

   +63+64+65+....+1008+1009 + 1010 - 63

S = 2·[(63+1009)+(64+1008)+...+(535+537) + 536] + 1010 - 63

S = 2·(1072+1072+1072+...+1072) {de 473 ori} + 2·536 + 947

S = 2·1072·473 + 2019

S = 1016131

Metoda II:

S = 127+129+131+...+2017+2019

S = (2·1+125)+(2·2+125)+(2·3+125)+...+(2·947+125)

{ (2019 - 125) : 2 = 947  (de aici l-am scos pe 947) }

S = 2·(1+2+3+...+946+947) + 125+125+125+...+125 {de 947 ori}

S = 2·(1+2+3+...+946+947) + 125·947

S = 2·[947·(947+1)]/2 + 125·947

S = 947·948 + 125·947

S = 947·(948+125)

S = 947·1073

⇒ S = 1016131

Metoda III:

$S = 127+129+131+...+2017+2019 \\\\S = 1+3+5+...+2019 - (1+3+5+...+125) \\\\S = \left(\dfrac{2019+1}{2}\right)^2 - \left(\dfrac{125+1}{2}\right)^2 \\\\S = \left(\dfrac{2020}{2}\right)^2 - \left(\dfrac{126}{2}\right)^2\\\\S = 1010^2 - 63^2\\\\\Rightarrow \boxed{S = 1016131}$