Question

Ajutor!
Calculați!
7+14+21+...+700
1+2+3+...+243
2+4+6+...+2016
5+10+15+20+...+250
17+18+...+239
37+38+...+100
1+4+7+10+...+301
1+5+9+13+...+401
2+5+8+11+...+107
Va multu!mesc!

Answer 1

7+14+21+...+700=
=7(1+2+3+...+100)=
=7*100*101/2=707*50=35350

1+2+3+...+243=
243(243+1) :2=
243*244 /2 =
59292 / 2 = 29646

2+4+6+...+2016 = 
2(1+2+3+---+1008)=
2*1008*1009 / 2 =2016 * 1009 / 2 =2034144 / 2 = 1017072

5+10+15+20+...+250
5(1+2+3+4+..+50)=
5*50*51/2 = 250*51 / 2= 12750/2 = 6375

17+18+...+239=
239(239+1)/2 = 
239*240 /2 = 
57360 / 2 =  28680

37+38+...+100 =
100(100+1) /2 =
100*101 / 2= 10100 / 2 = 5050

S =1+4+7+10+...+292+295+298+301
S = 301+298+295+292+...+10+7+4+1
__________________________________
2S = 302+302+302+302+...+302+302+302+302
stabilim cati termeni are suma
1, 1+3*1, 1+3*2, .........1+3*100
deci suma are 101 termeni
2S = 302*101
2S = 30502
S = 30502 /2
S = 15251

1+5+9+13+...+401
S = 1+5+9+13+.......389+393+397+401
S= 401+397+393+289+....13+9+5+1
__________________________________
2S = 402+402+402+........+402+402+402+402
1, 1+4*1, 1+4*2, 1+4*3..........1+4*100
2 S = 402*101
2S = 40602
S = 40602/2
S = 20301

2+5+8+11+...+107
S = 2+5+8+11+...98+101+104+107
S = 107+104+101+98+..........+11+8+5+2
_____________________________________
2S = 109+109+109+109+.........+109+109+109+109
2, 2+3*1,2+3*2,2+3*3..................2+3*35
2S = 109*36
2S = 3924
S = 3924/2
S = 1962