Question

Determinati  numerele  naturale  n  astfel  incat:

a)   3/n+1 _>sau=  1/2
b)   6/2n-3<sau= 1/5
c)    n+5/n+4>sau= 4/5
d)    2n+1  /4>3/4 si n <7
e)     n²/8< 20/4
f)     n³-1  /3<24/2 si n >sau= 2.

Answer 1

a) 3(n+1)≥1/2
3n+3≥1/2
2(3n+3)≥1
6n+6≥1
6n≥1-6
6n≥-5
n≥-5/6
n∈[-5/6,+α)

b)6/(2n-3)≤1/5
6x5≤2n-3
-2n≤-3-30
-2n≤-33       ·(-1)
n≥33/2
n∈[33/2,+α)

c)(n+5)/(n+4)≥4/5
5(n+5)≥4(n+4)
5n+5≥4n+4
5n-4n≥4-5
n≥-1
n∈[-1,+α)

d)(2n+1)/4>3/4
4(2n+1)>4x3
8n+4>12
8n>12-4
8n>8
n>8/8
n>1   si n<7
n∈(1,7)

e)n²/8<20/4
n²/8<5
n²<5x8
n²<40
n<√40
n<2√10
n∈(-α,2√10)

f) (n³-1)/3<24/2
(n³-1)/3<12
n³-1<12x3
n³<36+1
n³<37
n<∛37 si n≥2
n∈[2,∛37)