Question

1. Efectuati:
a) 7 + 9 + 11 + ... + 2013 - 5 - 7 - 9 - ... - 2011

b) 11 + 12 + 13 + ... + 78

c) 5 + 10 + 15 + ... + 95

d) 5 + 8 + 11 + ... + 305


2. Stiind ca x + 4y + 2z = 13 si 3x + 2z = 11, determinati x + y + z.

Answer 1

1.
a) 7 + 9 + 11 + ... + 2013 - 5 - 7 - 9 - ... - 2011=
=(7+9+11+...2011)+2013 - 5 - (7+9+11+...+2011)=
=2013 - 5 = 2008

b) 11 + 12 + 13 + ... + 78 = (78+11)*(78-11+1):2=89*68:2 =3026

c) 5 + 10 + 15 + ... + 95 =5*(1+2+3+...+19)=5*20*19:2=950

d) Suma =a+(a+k)+(a+2k)+...+n=(n+a)[(n-a):k+1]:2

5 + 8 + 11 + ... + 305 = (305+5)[(305-5):3+1]:2= 310*101:2=15655


2. 
x + 4y + 2z = 13
3x + 2z = 11,
Le adunam :
x+4y+2z+3x+2z=13+11
4x+4y+4z=24
4*(x+y+z)=24
z+x+y+z=6