Question

Calculati urmatoarele sume :

a) 1+2+3+.....+80=
b) 2+4+6.....+100=
c) 1+3+5...+99=
d) 3+7+11+15+.....+43=

Answer 1

a) 1 + 2 + 3 + ..........+ 80 = 80 * (80 + 1)/2 = 40 * 81 = 3240
b) 2 + 4 + 6 + .......+ 100 = 2 * 50 * (50 + 1)/2 = 2 * 51 * 25 = 2550
c) 1 + 3 + 5 + ........ + 99 = (99 + 1) * [(99 - 1)/2 + 1] : 2 = 100 * [49 + 1] : 2 =
    100 * 25 = 2500
d) 3 + 7 + 11 + 15 + ...... + 43 = (43 + 3) * [(43 - 3) : 4 + 1] : 2 = 46 * [40 : 4 + 1]     : 2 = 46 * 11 : 2 = 23 * 11 = 253

Answer 2

[tex]a)~S=~1+~2+3+~....+~80=\\\\ S= \frac{[80(80+1)]}{2} \\\\ S=40*81\\\\ S=3240 [/tex]

[tex]b)~S=~2~+~4~+6~+~...+100\\\\ S=2(1+2+3+...+50)\\\\ S= 2\frac{[50(50+1)]}{2} \\\\ S=50*51\\\\ S=2550[/tex]


[tex]c)~S=~1+3+5+...+99\\\\ 1=2*0+1\\ 3=2*1+1\\ 5=2*2+1\\ .........\\ 99=2*49+1\\\\ S=(2*0+1)+(2*1+1)+(2*2+ 1)+...+(2*49+1)\\ S=2*0+2*1+2*2+...+2*49+1+1+1+...+1\\ S=0+2(1+2+3+...+49)+50\\ S=2(49*50):2+50\\ S=49*50+50\\ S=2450+50\\ S=2500[/tex]