Question

Progresia geometrica (bn) n> sau egal decât 1 de rație q este definita prin anumite elemente date. Determinati in fiecare din cazuri elementele cerute

​

Answer 1

Răspuns:

Explicație pas cu pas:

Progresii geometrice

1) b₁ = 2 ; q = -3 ; b₆ = ?

b₆ = b₁·q⁵ = 2·(-3)⁵ = -486

2) b₁ = √2 ; b₂ = √8 ; b₁₀ = ?

b₂ = b₁ ·q => q = b₂:b₁ = √8 : √2 = √(8:2) = √4 = > q = 2

b₁₀ = b₁·q⁹ = √2·2⁹ = 512√2

3) b₁ = -1/2 ; b₈ = -2⁶ ; q = ?

b₈ = b₁·q⁷  => q⁷ = b₈/b₁ = (-2⁶):(-1/2) = 2⁷ => q = 2

4) b₅ = 61 ; b₁₁ = 1647 ; b₇ = ?

b₇ = b₅·q²  = b₁₁:q⁴ => q⁶ = 1647/61 = 27 => q= 27¹⁽⁶ = (3³)¹⁾⁶ = 3³⁾⁶ = 3¹⁾²=√3

q = √3 => b₇ = 61·3 => b₇ = 183

5) b₁-b₂ = 8 ; b₂+b₃ = 12 ; b₁ = ? q = ?

b₁-b₁·q = 8 <=> b₁(1-q) = 8

b₁·q + b₁·q² = 12 <=>b₁·q(1+q) = 12

b₁ = 8/(1-q)

8q(1+q) = 12(1-q)

8q+8q²=12-12q

8q²+20q-12 = 0  I:4

2q²+5q-3 = 0  => q₁,₂ = (-5±√(25+24)/4 = (-5±7)/4

q₁ = -12/4 => q₁ = -3 => b₁ = 8/4 = > b₁ = 2

q₂ = 2/4 => q₁ = 1/2 => b₁ = 16

#copaceibrainly