Question

Ex 34, 35(a,b) si 36!
Va rog mult°

Answer 1

34)
a)
(a+2b)/(a+b) +(b+2a)/(a+b)=(a+2b+b+2a)/(a+b)=(3a+3b)/(a+b)=3(a+b)/a+b)=3
b)
(3a+2b)/(a+b+c) +(3b+2c)/(a+b+c)+(3c+2a)/(a+b+c)=
=(3a+2b +3b+2c+3c+2a)/(a+b+c)=5a+5b+5c)/(a+b+c)=5(a+b+c)/(a+b+c+)=5



35.
1/6+1/7+1/8+....+1/30<5
avem 30-6+1=25 fractii
fractiile (1/7,1/8,....1/30) sunt <1/6
1/6+1/7+1/8+....+1/30<1/6+1/6+1/6+....+1/6
1/6+1/7+1/8+....+1/30<1/6×25
1/6+1/7+1/8+....+1/30<25/6     25/6 =4,1(6)            5>25/6
1/6+1/7+1/8+....+1/30<5

1/6+1/7+1/8+....+1/30>5/6
avem 30-6+1=25 fractii
e la 1/6 la 1/30 sunt 30-6+1=25 de fractii
fractiile (1/6, 1/7,1/8...1/29) sunt fiecare >1/30
rezulta :
 1/6+1/7+...+1/30>1/30+1/30+...+1/30
 1/6+1/7+...+1/30>1/30 ×25
 1/6+1/7+...+1/30>5/6

36.
      ____   ____    ____
S=[a,b(c)+b,c(a)+c,a(b)]/(a+b+c)
____
a,b(c)  =(100a+10b+c-10a-b)/90=(90a+9b+c)/90
____
b,c(a)  =(100b+10c+a-10b-c)/90=(90b+9c+a)/90
____
c,a(b)  =(100c+10b+b-10c-a)/90=(90c+9a+b)/90

      ____   ____    ____
S=[a,b(c)+b,c(a)+c,a(b)]/(a+b+c)=
=[(90a+9b+c)/90+(90b+9c+a)/90+(90c+9a+b)/90]/(a+b+c)
=(90a+9b+c+90b+9c+a+90c+9a+b)/90(a+b+c)=
=(100a+100b+100c)/90(a+b+c)=
=100(a+b+c)/90(a+b+c)=10/9=1,(1)